Ring, ring homomorphism, and ideal

Intuition Beginner

A ring is a number system with two operations that work together: an addition that lets you add and subtract, and a multiplication that lets you combine. The integers are the prototype. You can add two integers, subtract them, multiply them, and you always land back among the integers.

A ring is more than a group. A group has one operation. A ring has two, and they fit together by a distributive rule: multiplying a sum equals summing the products. The integers, rational numbers, real numbers, complex numbers, polynomials with real coefficients, and square matrices with integer entries are all rings.

A ring homomorphism is a map between two rings that respects both operations. It turns sums into sums and products into products. The map sending an integer to its remainder after dividing by $5$ is a ring homomorphism from the integers to the five-element clock system.

An ideal is a special kind of subset of a ring. It is closed under addition and subtraction inside itself, and it absorbs multiplication by anything from the larger ring. The even integers form an ideal inside the integers because adding two evens gives an even, and multiplying any integer by an even gives an even.

Why does this concept exist? Rings and ideals are the algebraic language for arithmetic with two operations, and they underpin everything from solving polynomial equations to the geometry of algebraic varieties.

Visual Beginner

The picture shows the integers arranged on a horizontal line, with the even integers marked in a second color. The evens form a sub-pattern that is closed under addition and is also absorbed by multiplication from outside.

Multiplying any integer by an even integer keeps you among the evens. This absorption property is what makes the evens an ideal.

Worked example Beginner

Work with the integers and the ideal $2\mathbb{Z}$ of even integers. Compute in the quotient $\mathbb{Z}/2\mathbb{Z}$, which has just two elements: even and odd.

Step 1. Add $3$ and $4$. The sum is $7$. In the quotient $\mathbb{Z}/2\mathbb{Z}$, the element $3$ is odd and $4$ is even. Odd plus even is odd. The sum $7$ is odd, agreeing with the quotient.

Step 2. Multiply $3$ and $4$. The product is $12$. In the quotient, odd times even is even. The product $12$ is even, agreeing with the quotient.

Step 3. The map $\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$ sending each integer to "even" or "odd" is a ring homomorphism. The kernel is exactly the even integers, which is the ideal $2\mathbb{Z}$.

What this tells us: an ideal lets you collapse a ring into a smaller ring, and the homomorphism that does the collapsing has the ideal as its kernel.

Check your understanding Beginner

Formal definition Intermediate+

A ring is a triple $(R,+,\cdot )$ where $R$ is a set and $+,\cdot :R\times R\to R$ are binary operations satisfying:

1. $(R,+)$ is an abelian group with identity $0$.
2. Multiplication is associative: $(ab)c=a(bc)$ for all $a,b,c\in R$.
3. There is a multiplicative identity $1\in R$ with $1\cdot a=a\cdot 1=a$ for all $a\in R$.
4. Multiplication distributes over addition on both sides: $a(b+c)=ab+ac$ and $(b+c)a=ba+ca$ for all $a,b,c\in R$.

A ring is commutative if $ab=ba$ for all $a,b\in R$. Throughout the rest of this unit, "ring" means "commutative ring" unless otherwise stated ^{[Lang §II.1]}.

A subring $S\subseteq R$ is a subset containing $1$ and closed under $+$, $-$, and $\cdot$. A ring homomorphism $\phi :R\to S$ is a function satisfying $\phi (a+b)=\phi (a)+\phi (b)$, $\phi (ab)=\phi (a)\phi (b)$, and $\phi (1_R)=1_S$.

An ideal of $R$ is a subset $I\subseteq R$ such that $(I,+)$ is a subgroup of $(R,+)$ and $r\cdot i\in I$ for all $r\in R$ and $i\in I$. The condition $r\cdot i\in I$ is the absorption property: an ideal swallows multiplication by any ring element. For non-commutative rings, one distinguishes left ideals (closed under left multiplication), right ideals (closed under right multiplication), and two-sided ideals (closed under both); only two-sided ideals support the quotient construction below ^{[Dummit-Foote §7.3]}.

The absorption property distinguishes an ideal from a mere subring. Every ideal $I\ne R$ misses $1$ (otherwise $r=r\cdot 1\in I$ for every $r$, making $I=R$), so proper ideals do not contain the multiplicative identity and are not subrings in the standard sense. The improper ideal $R$ itself is included as an ideal for technical reasons, allowing the ideal-theoretic lattice $\{(0),\dots ,R\}$ to have both least and greatest elements. The improper ideal corresponds geometrically to the empty subscheme of $\mathrm{Spec}(R)$, and the zero ideal $(0)$ corresponds to the entire scheme.

The kernel of a ring homomorphism is $\ker \phi =\{r\in R:\phi (r)=0\}$, which is always an ideal. The image $\mathrm{im}\phi =\{\phi (r):r\in R\}$ is a subring of $S$.

Given an ideal $I\subseteq R$, the quotient ring $R/I$ has elements the cosets $r+I$, with operations $(r+I)+(s+I)=(r+s)+I$ and $(r+I)(s+I)=rs+I$. Both operations are well-defined because $I$ absorbs multiplication. The map $R\to R/I$ sending $r\mapsto r+I$ is a surjective ring homomorphism with kernel $I$ ^{[Atiyah-Macdonald §1]}.

Verifying well-definedness of multiplication in the quotient: if $r+I=r^{\prime }+I$ and $s+I=s^{\prime }+I$, then $r-r^{\prime }\in I$ and $s-s^{\prime }\in I$. Compute $rs-r^{\prime }{s}^{\prime }=r(s-s^{\prime })+(r-r^{\prime })s^{\prime }$. The first term lies in $I$ by absorption (since $s-s^{\prime }\in I$ and $r\in R$); the second lies in $I$ for the same reason (since $r-r^{\prime }\in I$ and $s^{\prime }\in R$). Hence $rs-r^{\prime }{s}^{\prime }\in I$, so $rs+I=r^{\prime }{s}^{\prime }+I$. The proof uses ring multiplication's two-sided absorption property in an essential way, which is why two-sided ideals are required for the quotient construction in the non-commutative case.

A principal ideal $(a)=\{ra:r\in R\}$ is the smallest ideal containing $a$. More generally, given a set $S\subseteq R$, the ideal generated by $S$, denoted $(S)$, is the set of all finite $R$-linear combinations $r_1{s}_1+\cdots +r_n{s}_n$ with $r_i\in R$ and $s_i\in S$. An ideal is finitely generated if it has the form $(s_1,\dots ,s_n)$ for finitely many generators. The polynomial-ring ideal $(x_1,x_2)\subset k[x_1,x_2]$ requires two generators because no single polynomial divides both $x_1$ and $x_2$.

An ideal $\mathfrak{p}$ is prime if $\mathfrak{p}\ne R$ and $ab\in \mathfrak{p}$ implies $a\in \mathfrak{p}$ or $b\in \mathfrak{p}$. Equivalently, $\mathfrak{p}$ is prime if and only if $R/\mathfrak{p}$ is an integral domain (a commutative ring with $1\ne 0$ and no zero divisors). An ideal $\mathfrak{m}$ is maximal if $\mathfrak{m}\ne R$ and no ideal strictly between $\mathfrak{m}$ and $R$ exists. Equivalently, $\mathfrak{m}$ is maximal if and only if $R/\mathfrak{m}$ is a field. Every maximal ideal is prime, but not conversely: in $\mathbb{Z}$, the ideal $(0)$ is prime but not maximal, since $\mathbb{Z}/(0)=\mathbb{Z}$ is a domain but not a field.

The lattice of ideals of $R$ has natural operations. Given ideals $I,J\subseteq R$, the sum $I+J=\{a+b:a\in I,b\in J\}$ is the smallest ideal containing both; the intersection $I\cap J$ is again an ideal; the product $IJ$ is the ideal generated by all products $ab$ with $a\in I$ and $b\in J$. In general $IJ\subseteq I\cap J$ with equality when $I+J=R$ (the coprime case). The radical $\sqrt{I}=\{r\in R:r^n\in I\text{ for some }n\ge 1\}$ is an ideal; an ideal equal to its own radical is called a radical ideal, and these are exactly the intersections of prime ideals containing $I$ (Krull's theorem on the radical).

Examples Intermediate+

Several rings and their ideal structures form the working library of commutative algebra.

The integers $\mathbb{Z}$ form the prototype commutative ring with $1$. Every ideal of $\mathbb{Z}$ is principal: $I=(n)=n\mathbb{Z}$ for a unique $n\ge 0$, established by applying the division algorithm to the smallest positive element of $I$. The prime ideals are $(0)$ and $(p)$ for each prime $p$; the maximal ideals are exactly $(p)$, since $\mathbb{Z}/(p)={\mathbb{F}}_p$ is a field while $\mathbb{Z}/(0)=\mathbb{Z}$ is only a domain. Ideal operations are computed via gcd and lcm: $(m)+(n)=(\gcd (m,n))$, $(m)\cap (n)=(\mathrm{lcm}(m,n))$, $(m)(n)=(mn)$, and $\sqrt{(n)}=(\mathrm{rad}(n))$ where $\mathrm{rad}(n)$ is the product of distinct prime divisors of $n$.

The polynomial ring $k[x]$ over a field $k$ is also a principal ideal domain. The Euclidean division algorithm — divide $f(x)$ by $g(x)$ to obtain $f=qg+r$ with $\mathrm{deg}r<\mathrm{deg}g$ — drives the structural parallel with $\mathbb{Z}$. Every ideal $I\subseteq k[x]$ has the form $(p(x))$ for a unique monic generator $p(x)$, and the maximal ideals are exactly $(p(x))$ for $p$ irreducible. The prime ideals are $(0)$ and the $(p(x))$ for $p$ irreducible. When $k=\mathbb{C}$, the irreducible polynomials are exactly the linear $x-a$, so the maximal ideals of $\mathbb{C}[x]$ are $(x-a)$ for $a\in \mathbb{C}$, in bijection with $\mathbb{C}$. When $k=\mathbb{R}$, the irreducibles are $x-a$ and $x^2+bx+c$ with discriminant $b^2-4c<0$, giving the maximal ideals.

The polynomial ring $k[x_1,\dots ,x_n]$ in several variables over a field is no longer a principal ideal domain when $n\ge 2$. The ideal $(x_1,x_2)\subset k[x_1,x_2]$ has no single generator. It is, however, finitely generated, and more generally every ideal of $k[x_1,\dots ,x_n]$ is finitely generated by the Hilbert basis theorem (Theorem 2 at Master tier).

The matrix ring $M_n(R)$ over a commutative ring $R$ is non-commutative for $n\ge 2$. Its two-sided ideals are exactly the sets $M_n(I)$ for ideals $I\subseteq R$, while the left ideals are far more numerous (e.g., the set of matrices with prescribed support in a column). The non-commutative case is the Wedderburn-Artin context discussed at Master tier.

The ring of continuous real-valued functions $C(X,\mathbb{R})$ on a compact Hausdorff space $X$ provides a topological example. By Gelfand's theorem, the maximal ideals of $C(X,\mathbb{R})$ are exactly the ideals ${\mathfrak{m}}_p=\{f:f(p)=0\}$ for $p\in X$. The ring records the topology of $X$: maximal-ideal space recovers $X$, and homomorphisms $C(Y,\mathbb{R})\to C(X,\mathbb{R})$ correspond to continuous maps $X\to Y$.

The ring of $p$-adic integers ${\mathbb{Z}}_p={\underleftarrow{\lim }}_n\mathbb{Z}/p^n\mathbb{Z}$ is a complete local Noetherian ring with maximal ideal $(p)$ and residue field ${\mathbb{F}}_p$. Its ideals are exactly $(0)$ and $(p^n)$ for $n\ge 0$, giving a totally ordered ideal lattice. The localisation ${\mathbb{Q}}_p={\mathbb{Z}}_p[1/p]$ is the field of $p$-adic numbers. Modules over ${\mathbb{Z}}_p$ are the natural setting for $p$-adic representation theory of Galois groups, and the Cohen structure theorem identifies ${\mathbb{Z}}_p$ as the Cohen ring $C({\mathbb{F}}_p)$.

The Boolean ring $\mathcal{P}(X)$ of subsets of a set $X$, with symmetric difference as addition and intersection as multiplication, is a commutative ring in which every element is idempotent ($a^2=a$). Every prime ideal is maximal, and $\mathrm{Spec}(\mathcal{P}(X))$ is the Stone space of ultrafilters on $X$. Stone's representation theorem (Stone 1936) identifies every Boolean ring with the ring of clopen subsets of a totally disconnected compact Hausdorff space, providing the algebraic counterpart of two-valued logic and a paradigm case where $\mathrm{Spec}$ has a purely topological description.

Counterexamples to common slips Intermediate+

• An ideal is not just a subring. The set $\mathbb{Z}\subset \mathbb{Q}$ is a subring of the rationals but not an ideal: multiplying $1\in \mathbb{Z}$ by $\frac{1}{2}\in \mathbb{Q}$ gives $\frac{1}{2}$, which lies outside $\mathbb{Z}$. The absorption property fails.
• A ring homomorphism need not be injective. The map $\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ has kernel $n\mathbb{Z}$ and is far from injective when $n>1$. The first isomorphism theorem (below) recovers injectivity by quotienting out the kernel.
• Prime does not imply maximal. In the polynomial ring $\mathbb{Z}[x]$, the principal ideal $(x)$ is prime (since $\mathbb{Z}[x]/(x)\cong \mathbb{Z}$ is a domain) but not maximal (since $\mathbb{Z}$ is not a field). The maximal ideals of $\mathbb{Z}[x]$ are the ideals $(p,f(x))$ with $p$ prime and $f(x)$ irreducible modulo $p$.

Key theorem with proof Intermediate+

Theorem (First isomorphism theorem for rings). Let $\phi :R\to S$ be a ring homomorphism. Then $\ker \phi$ is an ideal of $R$, the image $\mathrm{im}\phi$ is a subring of $S$, and the induced map

$$\bar{\phi }:R/\ker \phi ⟶\mathrm{im}\phi ,r+\ker \phi ⟼\phi (r)$$

is a ring isomorphism.

Proof. We first verify that $\ker \phi$ is an ideal. The kernel is closed under addition: if $a,b\in \ker \phi$ then $\phi (a+b)=\phi (a)+\phi (b)=0+0=0$. It is closed under additive inverses: $\phi (-a)=-\phi (a)=0$. It contains $0$ since $\phi (0)=0$. Hence $(\ker \phi ,+)$ is a subgroup of $(R,+)$.

For absorption, let $r\in R$ and $a\in \ker \phi$. Then $\phi (ra)=\phi (r)\phi (a)=\phi (r)\cdot 0=0$, so $ra\in \ker \phi$. The kernel is an ideal.

That $\mathrm{im}\phi$ is a subring of $S$ follows from preservation of $+$, $\cdot$, and $1$: $\phi (a)+\phi (b)=\phi (a+b)\in \mathrm{im}\phi$, $\phi (a)\phi (b)=\phi (ab)\in \mathrm{im}\phi$, $1_S=\phi (1_R)\in \mathrm{im}\phi$, and $-\phi (a)=\phi (-a)\in \mathrm{im}\phi$.

We now define $\bar{\phi }$. Set $\bar{\phi }(r+\ker \phi )=\phi (r)$. We check this is well-defined: if $r+\ker \phi =r^{\prime }+\ker \phi$, then $r-r^{\prime }\in \ker \phi$, so $\phi (r)-\phi (r^{\prime })=\phi (r-r^{\prime })=0$, giving $\phi (r)=\phi (r^{\prime })$.

The map $\bar{\phi }$ preserves addition and multiplication directly from the definition of quotient operations and the homomorphism property of $\phi$. It also preserves $1$: $\bar{\phi }(1+\ker \phi )=\phi (1_R)=1_S$.

Injectivity: if $\bar{\phi }(r+\ker \phi )=0$, then $\phi (r)=0$, so $r\in \ker \phi$, giving $r+\ker \phi =\ker \phi$, the zero element of the quotient.

Surjectivity onto $\mathrm{im}\phi$: every element of $\mathrm{im}\phi$ has the form $\phi (r)$ for some $r$, and $\bar{\phi }(r+\ker \phi )=\phi (r)$ realises it.

Hence $\bar{\phi }$ is a bijective ring homomorphism onto $\mathrm{im}\phi$, i.e., a ring isomorphism. $\square$

Bridge. The first isomorphism theorem builds toward the correspondence between ideals of $R/I$ and ideals of $R$ containing $I$, and appears again in 01.02.07 (PID + UFD + Euclidean domain) where the structure of ideals in special classes of rings sharpens to the divisor lattice. The foundational reason the theorem works is that the kernel of a ring homomorphism is precisely the absorption-closed subgroup that records the relations imposed by the homomorphism, and this is exactly the algebraic version of the projection-onto-image construction. The bridge is between the lattice of ideals containing $\ker \phi$ and the lattice of ideals of the image. Putting these together with the construction of $\mathrm{Spec}(R)$ at the Master tier, the theorem identifies algebraic homomorphisms with geometric inclusions, and the pattern generalises to modules via the analogous first isomorphism theorem for modules in 01.02.09.

Exercises Intermediate+

Advanced results Master

Theorem 1 (Krull's theorem: every commutative ring has a maximal ideal). Let $R$ be a commutative ring with $1\ne 0$. Then $R$ has at least one maximal ideal. The proof is a direct application of Zorn's lemma: the collection of proper ideals of $R$, ordered by inclusion, is non-empty (it contains $(0)$) and every chain has an upper bound (the union of the chain remains a proper ideal because $1$ cannot lie in any proper ideal). Every chain has an upper bound, so a maximal element exists. As a corollary, every proper ideal $I⊊R$ is contained in at least one maximal ideal, applied to the ring $R/I$ ^{[Atiyah-Macdonald §1]}.

The existence of a maximal ideal in every commutative ring is equivalent over Zermelo-Fraenkel set theory to the axiom of choice; Hodges 1979 J. London Math. Soc. 19 established this equivalence. For Noetherian rings, the maximality result can be proved without choice by repeatedly extracting a strictly increasing chain of proper ideals and noting the chain must terminate. Specific maximal ideals are concrete in the worked examples: in $\mathbb{Z}$ they are $(p)$ for prime $p$; in $k[x]$ they are $(p(x))$ for irreducible monic $p$; in the ring of continuous real-valued functions on $[0,1]$, every maximal ideal has the form $\{f:f(p)=0\}$ for some $p\in [0,1]$, by Gelfand-Kolmogorov 1939.

Theorem 2 (Hilbert basis theorem). Let $R$ be a commutative Noetherian ring (every ideal is finitely generated). Then the polynomial ring $R[x]$ is also Noetherian. By induction, $R[x_1,\dots ,x_n]$ is Noetherian whenever $R$ is. In particular, $k[x_1,\dots ,x_n]$ over a field is Noetherian: every ideal has a finite generating set. Hilbert 1890 Math. Ann. 36, 473 proved this result and used it to establish finite generation of invariant rings of classical groups, closing a problem left open by Gordan's 1868 computational method ^{[Hilbert 1890]}. The proof uses leading-coefficient ideals: if $I\subseteq R[x]$ has leading coefficients generating $J\subseteq R$ with $J=(a_1,\dots ,a_r)$, then $I$ is generated by finitely many lifts of these generators together with $I\cap R[x{]}_{\le d}$ for $d$ bounded by the maximum degree.

Gordan reportedly remarked of Hilbert's proof "this is not mathematics, this is theology", reflecting the contrast between his explicit computational method for binary forms and Hilbert's abstract existence proof via the ascending chain condition. The basis theorem is the structural foundation of computational algebraic geometry: Buchberger 1965 introduced Gröbner bases as algorithmic finite generating sets for ideals of $k[x_1,\dots ,x_n]$, making the Hilbert basis theorem effective. Every Gröbner basis can be computed in finite time from any generating set via Buchberger's algorithm. The result also propagates upward: power series rings $R[[x_1,\dots ,x_n]]$ over Noetherian rings are Noetherian by the analogous Cohen argument, and finitely generated algebras over a field are Noetherian, putting nearly every ring of interest in algebraic geometry into the Noetherian regime.

Theorem 3 (Nakayama's lemma). Let $R$ be a commutative ring, $M$ a finitely generated $R$-module, and $I\subseteq R$ an ideal contained in the Jacobson radical $J(R)={\bigcap }_{\mathfrak{m}\mathrm{maximal}}\mathfrak{m}$. If $IM=M$, then $M=0$. Equivalently, if $M$ is finitely generated and $N\subseteq M$ is a submodule with $M=N+IM$, then $M=N$. For a local ring $(R,\mathfrak{m})$ with residue field $k=R/\mathfrak{m}$, elements $x_1,\dots ,x_n\in M$ generate $M$ if and only if their images generate $M/\mathfrak{m}M$ as a $k$-vector space. This unit's role as a "lifting" tool from quotient-field information back to the module pervades commutative algebra ^{[Eisenbud §4]}.

The standard proof uses the Cayley-Hamilton trick: if $M$ is generated by $x_1,\dots ,x_n$ and $IM=M$, write each $x_i={\sum }_j{a}_{ij}{x}_j$ with $a_{ij}\in I$. Then $(I_n-A)\mathbf{x}=0$ where $A=(a_{ij})$. Multiplying by the adjugate $\mathrm{adj}(I_n-A)$ gives $\det (I_n-A)\cdot x_i=0$ for all $i$. Expanding the determinant: $\det (I_n-A)=1+(\text{terms in }I)$, which is a unit modulo the Jacobson radical, hence a unit in $R$ (any element of $1+J(R)$ is a unit). Therefore $x_i=0$ for all $i$, so $M=0$. The Cayley-Hamilton step is itself a special case of the determinantal trick that drives many lifting theorems in commutative algebra. Application: in algebraic geometry, the local structure of a coherent sheaf at a closed point $p$ is controlled by its restriction to the residue field at $p$, which is the geometric content of Nakayama. In Lie theory, it underlies the Levi decomposition; in number theory, it lifts splittings from ${\mathbb{F}}_p$ to ${\mathbb{Z}}_p$.

Theorem 4 (Krull's principal ideal theorem, Hauptidealsatz). Let $R$ be a Noetherian commutative ring and let $f\in R$ be a non-zero-divisor. Then every minimal prime ideal containing $(f)$ has height $1$, i.e., contains no chain of primes of length greater than $1$ below it. Krull 1928 S.-B. Heidelberg. Akad. Wiss. 7, 11 ^{[Krull 1928]} established this as the foundation of dimension theory: the Krull dimension of a Noetherian ring is the supremum of lengths of prime chains, and the Hauptidealsatz controls how a single equation cuts dimension. Generalises to the $n$-generator theorem: a minimal prime over an ideal generated by $n$ elements has height at most $n$.

The geometric content of the Hauptidealsatz is that a single non-vanishing equation $f=0$ defines a hypersurface of codimension $1$ in an irreducible algebraic variety, and no smaller codimension can occur unless the equation is itself a unit or zero divisor. The Krull dimension of $k[x_1,\dots ,x_n]$ is $n$, matching the geometric dimension of affine $n$-space. The Krull dimension of $\mathbb{Z}$ is $1$: the prime chains are $(0)\subset (p)$. The Krull dimension of $\mathbb{Z}[x]$ is $2$: the prime chains are $(0)\subset (p)\subset (p,f(x))$ with $f$ irreducible modulo $p$, and also $(0)\subset (f(x))\subset (p,f(x))$ if $f$ has content $1$. Auslander-Buchsbaum-Serre 1956-1957 sharpened the dimension theory by characterising regular local rings via finite global dimension, opening the door to the homological methods that drive modern algebraic geometry (Cohen-Macaulay rings, Gorenstein rings, etc.).

Theorem 5 (Wedderburn-Artin theorem). Every semisimple Artinian ring is isomorphic to a finite direct product of matrix rings $M_{n_i}(D_i)$ over division rings $D_i$. The decomposition is unique up to ordering. Wedderburn 1908 Proc. London Math. Soc. 6, 77 proved the field case ^{[Wedderburn 1908]}; Artin 1927 Abh. Math. Sem. Univ. Hamburg 5, 251 extended it to general division-ring coefficients. For a finite-dimensional algebra over an algebraically closed field, this reduces to a sum of matrix rings over the field, governing the representation theory of finite groups via the group ring decomposition.

The Wedderburn-Artin theorem is the structural foundation of representation theory. Maschke's theorem (Maschke 1898) shows that the group algebra $k[G]$ of a finite group $G$ over a field $k$ of characteristic not dividing $|G|$ is semisimple Artinian. Combined with Wedderburn-Artin, this gives $k[G]\cong {\prod }_i{M}_{n_i}(D_i)$, identifying the irreducible representations with the matrix factors. When $k=\mathbb{C}$ and $G$ is finite, all $D_i=\mathbb{C}$, and the formula $|G|={\sum }_i{n}_i^2$ records the dimensions of the irreducible complex representations. Brauer 1929 extended the theory to modular representations, where $k[G]$ fails to be semisimple, opening a separate strand of structure theory via the Brauer correspondence. The Brauer group $\mathrm{Br}(k)$ of a field $k$ classifies finite-dimensional central simple algebras over $k$ up to Morita equivalence; it is the second Galois cohomology $H^2(\mathrm{Gal}(\bar{k}/k),{\bar{k}}^{\times })$ by a theorem of Châtelet 1944 and Serre 1959.

Theorem 6 (Cohen structure theorem). Every complete local Noetherian ring containing a field is a quotient of a power series ring $k[[x_1,\dots ,x_n]]$ over its residue field. Cohen 1946 Trans. AMS 59, 54 ^{[Cohen 1946]} established this as a uniformisation result: complete local rings look like quotients of formal power series rings, just as Riemann surfaces look locally like discs. The mixed-characteristic case introduces Cohen rings as the substitute for $k$.

The Cohen structure theorem is the algebraic version of formal-neighbourhood uniformisation in analytic geometry: just as the formal-power-series neighbourhood of a point on a smooth complex variety is the power series ring $\mathbb{C}[[x_1,\dots ,x_n]]$, the formal-power-series neighbourhood of any point on a Noetherian scheme is a quotient of such a power series ring. The mixed-characteristic refinement (when the residue field is, say, ${\mathbb{F}}_p$ and the ring has characteristic $0$) introduces the Cohen ring $C(k)$, an absolutely unramified complete discrete valuation ring with residue field $k$; for $k={\mathbb{F}}_p$, $C(k)={\mathbb{Z}}_p$ (the $p$-adic integers). The full statement: every complete local Noetherian ring is a quotient of $C(k)[[x_1,\dots ,x_n]]$ for some $n$, identifying every formal-neighbourhood algebra with a quotient of a power series ring over a Cohen ring. This is the foundational result underlying the $p$-adic geometry of Tate, Berkovich, and Scholze.

Theorem 7 (Spec construction and Zariski topology). For a commutative ring $R$, the prime spectrum $\mathrm{Spec}(R)$ is the set of prime ideals of $R$. The Zariski topology has closed sets $V(I)=\{\mathfrak{p}\in \mathrm{Spec}(R):\mathfrak{p}\supseteq I\}$ for ideals $I$. Maximal ideals are the closed points; if $R$ is an integral domain, the zero ideal $(0)$ is the generic point. Grothendieck 1960 Publ. Math. IHES 4 ^{[Grothendieck 1960]} used this construction to define the affine scheme $\mathrm{Spec}(R)$ with its structure sheaf ${\mathcal{O}}_{\mathrm{Spec}(R)}$, providing the local model for arbitrary schemes. Spec $\mathbb{Z}$ has closed points $(p)$ for each prime $p$ and the generic point $(0)$. Spec $k[x]$ for $k$ algebraically closed has closed points $(x-a)$ for $a\in k$ and the generic point $(0)$, recovering the affine line ${\mathbb{A}}_k^1$ at the level of closed points and adding a generic point for the function field.

The Zariski topology on $\mathrm{Spec}(R)$ is rarely Hausdorff: distinct closed points cannot be separated when their corresponding maximal ideals are comparable. The topology satisfies $V(IJ)=V(I)\cup V(J)$ and $V({\sum }_{\alpha }{I}_{\alpha })={\bigcap }_{\alpha }V(I_{\alpha })$, so $V$ reverses the lattice structure between ideals (after radicalisation) and closed subsets. The basis of open sets consists of the principal open sets $D(f)=\mathrm{Spec}(R)\setminus V(f)=\{\mathfrak{p}:f\notin \mathfrak{p}\}$ for $f\in R$; these are quasi-compact (every open cover has a finite subcover) and identify naturally with $\mathrm{Spec}(R[1/f])$. Functoriality is direct: a ring homomorphism $\phi :R\to S$ induces a continuous map $\mathrm{Spec}(\phi ):\mathrm{Spec}(S)\to \mathrm{Spec}(R)$ sending $\mathfrak{q}\mapsto {\phi }^{-1}(\mathfrak{q})$, and the contravariant functor $\mathrm{Spec}:{\mathbf{CRing}}^{\mathrm{op}}\to {\mathbf{Sch}}_{\mathrm{aff}}$ is an equivalence of categories with the category of affine schemes.

Theorem 8 (Nullstellensatz and the maximal-ideal correspondence). Let $k$ be an algebraically closed field. Then every maximal ideal of $k[x_1,\dots ,x_n]$ has the form $(x_1-a_1,\dots ,x_n-a_n)$ for some $(a_1,\dots ,a_n)\in k^n$. This is Hilbert's Nullstellensatz (Hilbert 1893 Math. Ann. 42, 313 ^{[Hilbert 1893]}): the points of affine space $k^n$ are in bijection with the maximal ideals of the polynomial ring. The radical correspondence says that $I(V(J))=\sqrt{J}$ for any ideal $J$: the ideal of polynomials vanishing on the common zero set of $J$ is the radical of $J$. This is the foundational result identifying classical algebraic geometry with commutative algebra over fields.

Theorem 9 (Localisation and local rings). Let $R$ be a commutative ring and $S\subseteq R$ a multiplicatively closed subset containing $1$. The localisation $S^{-1}R$ is the ring of formal fractions $\{r/s:r\in R,s\in S\}$ modulo the equivalence $r/s\sim r^{\prime }/s^{\prime }$ iff $t(r{s}^{\prime }-r^{\prime }s)=0$ for some $t\in S$. The localisation $R_{\mathfrak{p}}:=(R\setminus \mathfrak{p}{)}^{-1}R$ at a prime ideal $\mathfrak{p}$ is a local ring: a ring with a unique maximal ideal $\mathfrak{p}{R}_{\mathfrak{p}}$. Localisation commutes with quotients, with finite intersections of ideals, and with the formation of $\mathrm{Spec}$: $\mathrm{Spec}(S^{-1}R)$ is the open subset of $\mathrm{Spec}(R)$ avoiding $V(S)$. Stalks of the structure sheaf of $\mathrm{Spec}(R)$ at $\mathfrak{p}$ are exactly $R_{\mathfrak{p}}$. The local-global principle says many statements about modules over $R$ are equivalent to their truth at every localisation $R_{\mathfrak{p}}$: $M=0$ iff $M_{\mathfrak{p}}=0$ for all prime $\mathfrak{p}$ ^{[Atiyah-Macdonald §3]}.

The universal property of localisation is that it inverts the elements of $S$: any ring homomorphism $\phi :R\to T$ sending every $s\in S$ to a unit of $T$ factors uniquely through $S^{-1}R$. The fraction field $\mathrm{Frac}(R)$ of an integral domain $R$ is the localisation at $S=R\setminus \{0\}$. Localisation of a module $M$ at $S$ gives $S^{-1}M=M{\otimes }_R{S}^{-1}R$, and the functor $M\mapsto S^{-1}M$ is exact, making localisation one of the rare exact functors in commutative algebra. The local rings $R_{\mathfrak{p}}$ encode the infinitesimal structure of $\mathrm{Spec}(R)$ at the prime $\mathfrak{p}$, and a ring is regular at $\mathfrak{p}$ — i.e., $\mathrm{Spec}(R)$ is smooth at the corresponding point — when the local ring $R_{\mathfrak{p}}$ is regular. Regular local rings are characterised by Auslander-Buchsbaum-Serre as Noetherian local rings of finite global dimension, equivalently those whose maximal ideal is generated by exactly $\dim R$ elements.

Theorem 10 (Primary decomposition and the Lasker-Noether theorem). Let $R$ be a Noetherian commutative ring and $I⊊R$ a proper ideal. Then $I$ has a primary decomposition $I={\mathfrak{q}}_1\cap \cdots \cap {\mathfrak{q}}_n$, where each ${\mathfrak{q}}_i$ is a primary ideal (meaning $ab\in {\mathfrak{q}}_i$ with $a\notin {\mathfrak{q}}_i$ implies $b^n\in {\mathfrak{q}}_i$ for some $n$). The set of associated primes $\sqrt{{\mathfrak{q}}_i}$ is independent of the decomposition and is exactly the set of primes of the form $\{\mathrm{Ann}(x):x\in R/I\}$ that are prime. Lasker 1905 proved the polynomial-ring case, and Noether 1921 ^{[Noether 1921]} gave the general Noetherian version; this is the structural generalisation of unique prime factorisation to ideals in arbitrary Noetherian rings.

For $\mathbb{Z}$, primary decomposition recovers prime factorisation: the ideal $(n)={\prod }_i(p_i^{e_i})$ is the intersection ${\bigcap }_i(p_i^{e_i})$ with $(p_i^{e_i})$ primary (associated prime $(p_i)$). For $k[x_1,x_2]$ over an algebraically closed field, primary decomposition expresses the ideal $I=I(V)$ of a closed subscheme $V$ as the intersection of primary components, one for each irreducible component plus embedded components for non-reduced behaviour. Embedded primes — associated primes properly contained in some other associated prime — record the scheme-theoretic memory of non-radical ideals: the ideal $(x^2,xy)=(x)\cap (x^2,xy,y^2)=(x)\cap (x^2,y)$ in $k[x,y]$ has associated primes $(x)$ and $(x,y)$, with the embedded prime $(x,y)$ recording the non-reduced point at the origin. Bourbaki Algèbre Commutative Ch. 4 develops the theory in full generality, including the uniqueness of associated primes and the non-uniqueness of primary components corresponding to embedded primes.

Synthesis. The ring-and-ideal framework is the foundational reason that commutative algebra and algebraic geometry sit on the same scaffolding. The central insight is that an ideal $I\subseteq R$ records a system of equations, the quotient $R/I$ records the algebra of functions on the variety those equations define, and prime ideals record points of that variety together with the generic points of its irreducible components.

Putting these together with the Spec construction, the maximal-ideal correspondence in Theorem 8 identifies the closed points of $\mathrm{Spec}k[x_1,\dots ,x_n]$ with the affine $n$-space $k^n$ over an algebraically closed field $k$, and this is exactly the bridge to 04.02.01 schemes: the affine scheme $\mathrm{Spec}(R)$ is the local model of any scheme, with the Zariski topology and structure sheaf manufactured from the ring of regular functions. The pattern generalises from commutative rings to schemes via gluing, to derived schemes via derived rings, and to non-commutative geometry via the spectrum of a $C^{\ast }$-algebra (Gelfand-Naimark) or the modules over a non-commutative ring (Connes). The Hilbert basis theorem and the Hauptidealsatz make Noetherian commutative algebra effective: every ideal is finitely generated, dimension is finite, and the chain conditions support induction. Nakayama's lemma identifies "modulo the maximal ideal" with the local picture at a point, generalising the Taylor-expansion intuition from analysis.

Full proof set Master

Proposition 1 (Ideal-correspondence theorem). Let $\phi :R\to S$ be a surjective ring homomorphism with kernel $K$. Then the map $I\mapsto \phi (I)$ is an inclusion-preserving bijection between ideals of $R$ containing $K$ and ideals of $S$, with inverse $J\mapsto {\phi }^{-1}(J)$. Under this correspondence, prime ideals match prime ideals, and maximal ideals match maximal ideals.

Proof. We first show the maps are well-defined. If $I\supseteq K$ is an ideal of $R$, then $\phi (I)$ is an ideal of $S$: it is closed under addition because $\phi$ preserves addition, and it absorbs multiplication because $S=\phi (R)$ acts on $\phi (I)$ via $\phi (r)\cdot \phi (a)=\phi (ra)\in \phi (I)$ for $a\in I$ and $r\in R$, using surjectivity to write every element of $S$ in the form $\phi (r)$. If $J\subseteq S$ is an ideal, then ${\phi }^{-1}(J)\subseteq R$ is an ideal containing $K$ (since $0\in J$).

Inverse property: ${\phi }^{-1}(\phi (I))=I$ when $I\supseteq K$. The inclusion $\subseteq$ uses $\phi (I)=\phi (I^{\prime })$ with $I^{\prime }={\phi }^{-1}(\phi (I))$ and the fact that $I^{\prime }\supseteq I$ together with the kernel containment $I\supseteq K$ forces $I^{\prime }\subseteq I+K=I$. The inclusion $\supseteq$ is direct.

Conversely, $\phi ({\phi }^{-1}(J))=J$ because $\phi$ is surjective: every $s\in J$ has a preimage $r\in R$ with $\phi (r)=s$, and $r\in {\phi }^{-1}(J)$.

Prime-and-maximal preservation: if $I$ is prime and $I\supseteq K$, then $R/I$ is a domain, and $R/I\cong S/\phi (I)$ via the induced quotient map, so $\phi (I)$ is prime. The maximal-ideal case is identical with "domain" replaced by "field". The reverse direction follows from the bijection and the fact that the correspondence preserves inclusions. $\square$

Proposition 2 (Maximal ideals of $k[x]$ for $k$ a field). Let $k$ be a field. Then the maximal ideals of $k[x]$ are precisely the ideals $(p(x))$ generated by irreducible monic polynomials.

Proof. The polynomial ring $k[x]$ is a Euclidean domain (with degree as the Euclidean norm), hence a principal ideal domain. Every ideal has the form $(f(x))$ for some $f(x)\in k[x]$. The ideal $(f(x))$ is maximal in $k[x]$ if and only if $k[x]/(f(x))$ is a field.

If $f(x)$ is reducible, say $f(x)=g(x)h(x)$ with $\mathrm{deg}g,\mathrm{deg}h\ge 1$, then in the quotient $\bar{g}(x)\cdot \bar{h}(x)=0$ with both factors non-zero, so the quotient has zero divisors and is not a field; hence $(f(x))$ is not maximal.

If $f(x)=p(x)$ is irreducible, we show $k[x]/(p(x))$ is a field. Let $\bar{g}\in k[x]/(p(x))$ be non-zero, i.e., $g(x)$ is not divisible by $p(x)$. Since $p(x)$ is irreducible and $g(x)\not\equiv 0(\mathrm{mod}p(x))$, the gcd of $p(x)$ and $g(x)$ in $k[x]$ is a unit. By the Euclidean property of $k[x]$, there exist $a(x),b(x)\in k[x]$ with $a(x)p(x)+b(x)g(x)=1$. Reducing modulo $p(x)$: $\bar{b}\cdot \bar{g}=1$ in $k[x]/(p(x))$, so $\bar{g}$ has an inverse. Hence the quotient is a field.

The maximal ideals of $k[x]$ are exactly the $(p(x))$ with $p(x)$ monic irreducible (every ideal has a unique monic generator). $\square$

Proposition 2.5 (Ideal-theoretic characterisations of fields). Let $R$ be a commutative ring with $1\ne 0$. The following are equivalent:

(i) $R$ is a field.

(ii) The only ideals of $R$ are $(0)$ and $R$.

(iii) Every non-zero ring homomorphism $R\to S$ (with $S$ commutative) is injective.

(iv) The zero ideal $(0)$ is maximal.

Proof. (i) $\Rightarrow$ (ii): if $I$ is a non-zero ideal, choose $a\in I$ with $a\ne 0$. Since $R$ is a field, $a^{-1}\in R$, so $1=a\cdot a^{-1}\in I$, giving $I=R$.

(ii) $\Rightarrow$ (iii): the kernel of a ring homomorphism is an ideal. If the only ideals are $(0)$ and $R$, then either the kernel is $(0)$ (injective) or the kernel is $R$ (the homomorphism is zero, ruled out by hypothesis).

(iii) $\Rightarrow$ (iv): if $I⊊R$ is an ideal strictly between $(0)$ and $R$, then the quotient map $R\to R/I$ is non-zero with non-zero kernel $I$, contradicting (iii). So $(0)$ is maximal.

(iv) $\Rightarrow$ (i): if $(0)$ is maximal, then $R/(0)=R$ is a field by the standard characterisation of maximal ideals.

The equivalence makes fields ideal-theoretically simple: a ring is a field exactly when its ideal lattice is the two-element lattice $\{(0),R\}$. This is the algebraic counterpart of the geometric statement that a point has a discrete one-element topology with no proper closed subsets. $\square$

Proposition 3 (Chinese remainder theorem for commutative rings). Let $R$ be a commutative ring and $I_1,\dots ,I_n\subseteq R$ pairwise coprime ideals (meaning $I_i+I_j=R$ for $i\ne j$). Then the natural map

$$R/(I_1\cap \cdots \cap I_n)⟶R/I_1\times \cdots \times R/I_n$$

is a ring isomorphism, and $I_1\cap \cdots \cap I_n=I_1{I}_2\cdots I_n$.

Proof. We prove both the isomorphism and the intersection-equals-product claim by induction on $n$.

For $n=2$: the map $\phi :R\to R/I_1\times R/I_2$ sending $r\mapsto (r+I_1,r+I_2)$ is a ring homomorphism with kernel $I_1\cap I_2$. For surjectivity, use coprimality: write $1=a_1+a_2$ with $a_1\in I_1$ and $a_2\in I_2$. Then for any target $(r_1+I_1,r_2+I_2)$, the element $r=r_1{a}_2+r_2{a}_1$ maps to it: $r=r_1(1-a_1)+r_2{a}_1\equiv r_1(\mathrm{mod}{I}_1)$ and $r=r_1{a}_2+r_2(1-a_2)\equiv r_2(\mathrm{mod}{I}_2)$. The first isomorphism theorem gives the ring isomorphism $R/(I_1\cap I_2)\cong R/I_1\times R/I_2$.

For the intersection-equals-product claim: $I_1{I}_2\subseteq I_1\cap I_2$ always holds. Conversely, take $x\in I_1\cap I_2$. Multiply the coprimality equation $1=a_1+a_2$ by $x$: $x=a_{1}x+a_{2}x$. Since $x\in I_2$ and $a_1\in I_1$, the term $a_{1}x\in I_1{I}_2$; since $x\in I_1$ and $a_2\in I_2$, the term $a_{2}x\in I_1{I}_2$. Hence $x\in I_1{I}_2$, so $I_1\cap I_2\subseteq I_1{I}_2$.

For the inductive step: set $J=I_2\cap \cdots \cap I_n$. The pairwise coprimality propagates: $I_1+J=R$ because $I_1+I_i=R$ for each $i=2,\dots ,n$, so we can write $1=a_i+b_i$ with $a_i\in I_1$ and $b_i\in I_i$, and the product $b_2{b}_3\cdots b_n\in J$ together with the expansion $\prod (a_i+b_i)=1$ shows $1\in I_1+J$. Apply the $n=2$ case to $I_1$ and $J$, then the inductive hypothesis to $J$. $\square$

Connections Master

• Group 01.02.01. The additive structure of a ring is an abelian group, and the ideals of a ring are precisely the additive subgroups that absorb multiplication by ring elements. This is exactly the ring-theoretic upgrade of the group-theoretic normal-subgroup-and-quotient construction: ideals are to rings what normal subgroups are to groups, and the first isomorphism theorem for rings parallels the first isomorphism theorem for groups proved in that unit.

• Field 01.01.01. A field is a commutative ring in which every non-zero element has a multiplicative inverse; equivalently, a commutative ring whose only ideals are $(0)$ and the whole ring. The maximal-ideal characterisation $R/\mathfrak{m}$ is a field provides the foundational bridge: residue fields of local rings, function fields of varieties, and number fields all arise as quotients of rings by maximal ideals.

• PID, UFD, Euclidean domain 01.02.07. Builds toward the classification of special classes of commutative rings by ideal-theoretic and divisibility properties. A Euclidean domain has a division algorithm; a principal ideal domain has every ideal generated by one element; a unique factorisation domain has unique prime factorisation. The chain Euclidean $\Rightarrow$ PID $\Rightarrow$ UFD holds in general, and the examples $\mathbb{Z}$, $k[x]$, and $\mathbb{Z}[i]$ from this unit's exercises become the canonical witnesses.

• Modules 01.02.09. Modules generalise the abelian-group concept to allow scalar action by a ring; abelian groups are $\mathbb{Z}$-modules, vector spaces are $k$-modules over a field $k$, and ideals are themselves submodules of $R$ viewed as a module over itself. The downstream specialisation in the modules unit develops free modules, tensor products, and the structure theorem for finitely generated modules over a PID.

• Scheme 04.02.01. Spec $R$ is the affine scheme attached to a commutative ring $R$, the gluing-block of algebraic geometry. The Spec construction outlined in Theorem 7 is the foundational bridge from commutative algebra to algebraic geometry: rings become spaces with structure sheaves, ring homomorphisms become morphisms of schemes, and the Nullstellensatz of Theorem 8 identifies the closed points of $\mathrm{Spec}k[x_1,\dots ,x_n]$ with the classical affine space $k^n$ for $k$ algebraically closed.

Historical & philosophical context Master

Richard Dedekind 1871 introduced ideals (originally "ideale Zahlen") in supplement XI to the fourth edition of Dirichlet's Vorlesungen über Zahlentheorie ^{[Dedekind 1871]}, rescuing unique factorisation in algebraic number rings such as $\mathbb{Z}[\sqrt{-5}]$ where ordinary elements fail to factor uniquely. Kummer's 1846 "ideal numbers" for cyclotomic fields were the precursor; Dedekind's reformulation in terms of subsets closed under addition and absorbing multiplication gave the modern axiomatic definition. David Hilbert 1890 Math. Ann. 36, 473 proved the basis theorem ^{[Hilbert 1890]}, establishing that the polynomial ring $k[x_1,\dots ,x_n]$ is Noetherian and closing Gordan's invariant-theory problem in a single conceptual stroke. Hilbert 1893 Math. Ann. 42, 313 proved the Nullstellensatz ^{[Hilbert 1893]}, identifying the points of affine space with the maximal ideals of the polynomial ring over an algebraically closed field.

Emmy Noether 1921 Math. Ann. 83, 24 Idealtheorie in Ringbereichen ^{[Noether 1921]} gave the modern abstract treatment of ideal theory and identified the ascending chain condition as the structural property unifying Hilbert's basis theorem with primary decomposition. Noether's lectures at Göttingen in the 1920s and her students (van der Waerden, Krull, Brauer) transmitted the abstract axiomatic approach that became Moderne Algebra (van der Waerden 1930) and the foundation of modern algebra textbooks. Wolfgang Krull 1928 S.-B. Heidelberg. Akad. Wiss. 7, 11 ^{[Krull 1928]} proved the Hauptidealsatz and developed dimension theory.

Joseph Wedderburn 1908 Proc. London Math. Soc. 6, 77 ^{[Wedderburn 1908]} classified semisimple algebras over fields, and Cohen 1946 Trans. AMS 59, 54 ^{[Cohen 1946]} proved the structure theorem for complete local Noetherian rings. Oscar Zariski 1937-1947 developed the Zariski topology on varieties and the local-ring techniques of algebraic geometry over arbitrary fields. Alexander Grothendieck 1960 Publ. Math. IHES 4, Éléments de géométrie algébrique I ^{[Grothendieck 1960]} introduced the scheme as $\mathrm{Spec}(R)$ with its structure sheaf, completing the unification of commutative algebra and algebraic geometry that this unit's framework rests on. The lineage from Dedekind to Grothendieck spans 90 years and replaces the implicit number-theoretic content of factorisation with an explicit geometric structure on the space of prime ideals.

The philosophical content of the Dedekind-Hilbert-Noether-Grothendieck arc is the replacement of computational manipulation by structural axiomatics. Dedekind's ideals removed the need to manipulate ideal numbers as fictitious entities; Hilbert's basis theorem replaced Gordan's explicit invariant generators with an existence proof from the ascending chain condition; Noether's Idealtheorie axiomatised the chain condition itself; Krull's dimension theory turned ring-theoretic data into geometric invariants; Grothendieck's scheme theory turned geometric varieties into ring-theoretic objects via $\mathrm{Spec}$. Each step traded explicit calculation for structural generality, and the result was a framework in which problems from number theory (Fermat's Last Theorem via modular forms and Galois representations), algebraic geometry (Mori's minimal model program), and physics (mirror symmetry, geometric Langlands) could be formulated and addressed using a single algebraic vocabulary.

Contemporary developments extend the ring-and-ideal framework in several directions. Non-commutative algebraic geometry (Connes 1985-, Artin-Zhang 1994, Kontsevich-Rosenberg 2004) treats non-commutative rings as the function rings of generalised spaces, with quantum tori and Sklyanin algebras as canonical examples. Derived algebraic geometry (Lurie 2009, Toën-Vezzosi 2008) replaces commutative rings with simplicial commutative rings or $E_{\infty }$-ring spectra, generalising the ideal-and-quotient framework to the derived category. Perfectoid geometry (Scholze 2012 Publ. Math. IHES 116, 245) introduces perfectoid rings as a bridge between characteristic-$p$ and characteristic-$0$ Galois representations, with perfectoid spaces playing the role of $\mathrm{Spec}$ for these rings. Tropical geometry (Mikhalkin 2005, Maclagan-Sturmfels 2015) replaces $\mathrm{Spec}(R)$ with a piecewise-linear shadow, while motivic homotopy theory (Voevodsky 1996, Morel 2012) treats algebraic varieties as objects of a derived category of motives. Each of these extensions takes the ring-and-ideal apparatus of this unit as its starting point.

Bibliography Master

@incollection{Dedekind1871,
  author = {Dedekind, Richard},
  title = {{\"U}ber die Theorie der ganzen algebraischen Zahlen},
  booktitle = {Vorlesungen {\"u}ber Zahlentheorie, 4. Auflage},
  editor = {Dirichlet, P. G. L.},
  publisher = {Vieweg},
  year = {1871},
  note = {Supplement XI}
}

@article{Hilbert1890,
  author = {Hilbert, David},
  title = {Ueber die Theorie der algebraischen Formen},
  journal = {Mathematische Annalen},
  volume = {36},
  year = {1890},
  pages = {473--534}
}

@article{Hilbert1893,
  author = {Hilbert, David},
  title = {Ueber die vollen Invariantensysteme},
  journal = {Mathematische Annalen},
  volume = {42},
  year = {1893},
  pages = {313--373}
}

@article{Noether1921,
  author = {Noether, Emmy},
  title = {Idealtheorie in Ringbereichen},
  journal = {Mathematische Annalen},
  volume = {83},
  year = {1921},
  pages = {24--66}
}

@article{Krull1928,
  author = {Krull, Wolfgang},
  title = {Primidealketten in allgemeinen Ringbereichen},
  journal = {Sitzungsberichte der Heidelberger Akademie der Wissenschaften},
  volume = {7},
  year = {1928},
  pages = {11--29}
}

@article{Wedderburn1908,
  author = {Wedderburn, Joseph H. M.},
  title = {On hypercomplex numbers},
  journal = {Proceedings of the London Mathematical Society},
  volume = {6},
  year = {1908},
  pages = {77--118}
}

@article{Cohen1946,
  author = {Cohen, Irvin S.},
  title = {On the structure and ideal theory of complete local rings},
  journal = {Transactions of the American Mathematical Society},
  volume = {59},
  year = {1946},
  pages = {54--106}
}

@article{Grothendieck1960,
  author = {Grothendieck, Alexander},
  title = {{\'E}l{\'e}ments de g{\'e}om{\'e}trie alg{\'e}brique I: Le langage des sch{\'e}mas},
  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume = {4},
  year = {1960},
  pages = {5--228}
}

@book{LangAlgebra,
  author = {Lang, Serge},
  title = {Algebra},
  publisher = {Springer},
  year = {2002},
  edition = {3rd},
  series = {GTM 211}
}

@book{AtiyahMacdonald,
  author = {Atiyah, Michael F. and Macdonald, Ian G.},
  title = {Introduction to Commutative Algebra},
  publisher = {Addison-Wesley},
  year = {1969}
}

@book{Eisenbud,
  author = {Eisenbud, David},
  title = {Commutative Algebra with a View Toward Algebraic Geometry},
  publisher = {Springer},
  year = {1995},
  series = {GTM 150}
}

@book{DummitFoote,
  author = {Dummit, David S. and Foote, Richard M.},
  title = {Abstract Algebra},
  publisher = {John Wiley \& Sons},
  year = {2004},
  edition = {3rd}
}

@book{Artin,
  author = {Artin, Michael},
  title = {Algebra},
  publisher = {Pearson},
  year = {2011},
  edition = {2nd}
}