01.02.14 · foundations / groups

Semisimple rings, the Artin-Wedderburn structure theorem, the Jacobson radical

shipped3 tiersLean: none

Anchor (Master): Wedderburn 1908 On hypercomplex numbers (Proc. LMS); Artin 1927 Zur Theorie der hyperkomplexen Zahlen; Jacobson 1945 Structure theory of simple rings; Rieffel 1965 density-theorem proof

Intuition Beginner

A ring is a number system where you can add, subtract, and multiply, though multiplication need not commute. Some rings are built out of clean, indivisible pieces; others have a hidden sticky layer that glues parts together in a tangled way. This unit is about telling the two situations apart.

A module is a space the ring acts on, the way matrices act on vectors. A module is called simple when it has no proper non-zero piece preserved by the action: you cannot break it into a smaller invariant part. Simple modules are the atoms.

A ring is called semisimple when every space it acts on is a clean stack of these atoms, with no tangling. The payoff is a complete classification: each semisimple ring is just a product of rings of square matrices over division systems. Nothing more complicated can occur.

When a ring is not semisimple, there is a measurable cause. A special set of elements, the radical, behaves like the part of the ring that acts as zero on every atom. Removing the radical always restores the clean picture.

Visual Beginner

Picture a tall stack of identical flat tiles. Each tile is one simple module, an atom of the theory. A semisimple ring is one whose every space looks like such a stack: separate, parallel tiles you can pull apart by hand with no glue between them.

Next to the clean stack, draw a block where a thin grey layer seeps between two tiles and bonds them. That grey layer is the radical. The whole structure theorem says: scrape off the grey layer and what remains always falls apart into neat stacks of matrix blocks.

Worked example Beginner

Take the ring of all $2 \times 2$ matrices with real entries, written $M_{2} (R)$ . We check that the space it most naturally acts on is a clean stack.

Step 1. The natural space is the set of column vectors $R^{2}$ . A matrix acts on a column by ordinary matrix-times-vector multiplication.

Step 2. Is $R^{2}$ an atom? A preserved piece would be a line through the origin sent into itself by every matrix. But rotation by ninety degrees sends every line off itself, so no line survives. This means $R^{2}$ has no proper preserved piece: it is a single atom.

Step 3. Now view the ring acting on itself. Each column slot of a $2 \times 2$ matrix is a separate copy of $R^{2}$ . So the ring, acting on itself, is a stack of two identical atoms.

The conclusion: $M_{2} (R)$ is semisimple, and the matrix shape you started with is exactly the shape the structure theorem promises. A full matrix ring over a division system is the simplest building block.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $R$ is a ring with identity (not assumed commutative), and modules are left $R$ -modules unless stated otherwise ^{[Lang Algebra Ch. XVII]}.

Definition (Simple module). A non-zero $R$ -module $S$ is simple if its only submodules are $0$ and $S$ .

Definition (Semisimple module). An $R$ -module $M$ is semisimple if it is a direct sum of simple submodules. Equivalently (proved below), every submodule of $M$ is a direct summand. The zero module is semisimple, as the empty direct sum.

Definition (Semisimple ring). A ring $R$ is semisimple if $R$ , regarded as a left module over itself, is a semisimple module. (One shows this left-right symmetric notion is equivalent to: every $R$ -module is semisimple.)

Definition (Jacobson radical). The Jacobson radical $J (R)$ is the intersection of all maximal left ideals of $R$ . Equivalently, $J (R) = ⋂_{S} Ann_{R} (S)$ , the intersection of annihilators of all simple left $R$ -modules, which exhibits $J (R)$ as a two-sided ideal. A third characterisation: $x \in J (R)$ if and only if $1 - r x$ has a left inverse for every $r \in R$ .

Definition (Division ring). A division ring $D$ is a ring in which every non-zero element has a two-sided multiplicative inverse. A field is a commutative division ring; the quaternions $H$ are a non-commutative example.

Counterexamples to common slips Intermediate+

Semisimple is not the same as simple. The ring $M_{2} (R) \times M_{3} (R)$ is semisimple but has a proper two-sided ideal, so it is not a simple ring. Semisimplicity allows several matrix blocks.
A radical can be non-zero without the ring being a domain. The ring of upper-triangular $2 \times 2$ matrices over a field has Jacobson radical equal to the strictly upper-triangular matrices, a non-zero nilpotent ideal. The ring is neither semisimple nor a domain.
Nilradical and Jacobson radical differ. For a commutative ring the nilradical (nilpotent elements) sits inside $J (R)$ but can be strictly smaller: the localisation $Z_{(p)}$ has zero nilradical yet Jacobson radical $p Z_{(p)}$ . In Lang's terminology these two radicals coincide only under finiteness hypotheses.

Key theorem with proof Intermediate+

Theorem (Equivalence of the semisimplicity conditions). For an $R$ -module $M$ the following are equivalent: (i) $M$ is a sum of simple submodules; (ii) $M$ is a direct sum of simple submodules; (iii) every submodule of $M$ is a direct summand.

Proof. (i) $\Rightarrow$ (ii). Write $M = \sum_{i \in I} S_{i}$ with each $S_{i}$ simple. By Zorn's lemma choose a subset $J \subseteq I$ maximal with respect to the property that the sum $\sum_{j \in J} S_{j}$ is direct. Set $N = ⨁_{j \in J} S_{j}$ . For any $i$ , the intersection $S_{i} \cap N$ is a submodule of the simple module $S_{i}$ , so it is $0$ or $S_{i}$ . If it were $0$ , then $S_{i} + N$ would be direct, contradicting maximality of $J$ . So $S_{i} \subseteq N$ for all $i$ , giving $M = N = ⨁_{j \in J} S_{j}$ .

(ii) $\Rightarrow$ (iii). Let $P \subseteq M$ be a submodule. By Zorn pick $J$ maximal with $P \cap ⨁_{j \in J} S_{j} = 0$ ; set $N = ⨁_{j \in J} S_{j}$ . For each $i$ , the module $S_{i} \cap (P + N)$ is $0$ or $S_{i}$ ; if it were $0$ then $P \cap (N \oplus S_{i}) = 0$ , contradicting maximality. Hence every $S_{i} \subseteq P + N$ , so $P + N = M$ , and the choice of $J$ gives $P \cap N = 0$ . Thus $M = P \oplus N$ and $P$ is a summand.

(iii) $\Rightarrow$ (i). First, every non-zero submodule of $M$ contains a simple submodule: take any $0 \neq = x$ , and inside the cyclic module $R x$ choose, by Zorn, a submodule $K$ maximal among those not containing $x$ ; then $R x / K$ is simple, and writing $M = K \oplus K^{'}$ and intersecting with $R x$ produces a simple summand. Now let $M_{0}$ be the sum of all simple submodules. Splitting $M = M_{0} \oplus M_{1}$ by (iii), if $M_{1} \neq = 0$ it would contain a simple submodule, which by definition lies in $M_{0} \cap M_{1} = 0$ , a contradiction. So $M_{1} = 0$ and $M = M_{0}$ is a sum of simples. $□$

Bridge. This equivalence builds toward the Artin-Wedderburn theorem proved at Master tier, where the direct-summand formulation is the foundational reason a semisimple ring splits into blocks: writing $R = ⨁ S_{i}$ as left ideals and grouping isomorphic simples is exactly the step that produces the matrix factors. The pattern generalises the linear-algebra fact that every subspace has a complement, and the central insight is that semisimplicity is precisely the ring-theoretic upgrade of "every subspace is a direct summand" from vector spaces to modules. Putting these together, the splitting result appears again in 07.02.01 (Maschke's theorem), where averaging over a finite group supplies the complement that makes $C [G]$ semisimple, and the bridge is that the abstract complement here and the group-averaged projection there are the same construction seen at two altitudes.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that $J (R)$ , defined as the intersection of all maximal left ideals, annihilates every simple left $R$ -module, and conclude $J (R)$ is a two-sided ideal.

Hint

A simple module is $R / m$ for a maximal left ideal $m$ ; an element killing every $R / m$ lies in every $m$ .

Answer

Let $S$ be a simple left module and $0 \neq = s \in S$ . The map $r \mapsto r s$ is surjective with kernel a maximal left ideal $m_{s}$ , so $S ≅ R / m_{s}$ . If $x \in J (R)$ then $x \in m_{s}$ , so $x s = 0$ ; as $s$ was arbitrary, $x$ annihilates $S$ . Thus $J (R) \subseteq ⋂_{S} Ann_{R} (S)$ . Conversely every maximal left ideal is the annihilator of a single element of some simple $R / m$ , giving the reverse inclusion, so $J (R) = ⋂_{S} Ann_{R} (S)$ . Each annihilator $Ann_{R} (S) = {x : x S = 0}$ is two-sided because $x S = 0$ implies $(x r) S \subseteq x S = 0$ and $(r x) S = r (x S) = 0$ . An intersection of two-sided ideals is two-sided.

Exercise 4 (medium, symbolic).

Prove that $R$ is semisimple if and only if $J (R) = 0$ and $R$ is left-Artinian.

Hint

For one direction, a semisimple ring is a finite direct sum of simple left ideals, hence Artinian, and its radical kills every simple summand. For the converse, use that $R / J (R)$ is semisimple for Artinian $R$ .

Answer

If $R$ is semisimple, then $R = ⨁_{i = 1}^{n} L_{i}$ is a finite direct sum of simple left ideals (finite because $1$ lies in a finite partial sum), so $R$ is left-Artinian. Each $L_{i}$ is a simple module, and $J (R)$ annihilates it; since $R = \sum L_{i}$ and $J (R) \cdot L_{i} = 0$ , we get $J (R) = J (R) \cdot R = 0$ .

Conversely, suppose $R$ is left-Artinian with $J (R) = 0$ . For Artinian $R$ the quotient $R / J (R)$ is semisimple (proved at Master tier via nilpotence of the radical and lifting idempotents). With $J (R) = 0$ this says $R = R / J (R)$ is semisimple.

Exercise 7 (hard, symbolic).

Let $D$ be a division ring and $n \geq 1$ . Show $M_{n} (D)$ is simple (its only two-sided ideals are $0$ and the whole ring) and that its unique simple left module is $D^{n}$ , with $End_{M_{n} (D)} (D^{n}) ≅ D^{op}$ .

Hint

Use matrix units $e_{ij}$ : a non-zero two-sided ideal containing a matrix with entry $a \neq = 0$ contains $e_{k i} (a e_{ij}) e_{j l} = a e_{k l}$ , hence all of $M_{n} (D)$ .

Answer

Let $I \neq = 0$ be a two-sided ideal and $0 \neq = A \in I$ with $(i, j)$ -entry $a \neq = 0$ . Then $e_{k i} A e_{j l} = a e_{k l} \in I$ for all $k, l$ ; multiplying by $a^{- 1} e_{k k}$ on the left yields $e_{k l} \in I$ , so $I$ contains all matrix units and $I = M_{n} (D)$ . Hence $M_{n} (D)$ is simple.

The column space $D^{n}$ is a left $M_{n} (D)$ -module. It is simple: given $0 \neq = v$ , suitable matrices move $v$ to any chosen basis column, so $M_{n} (D) v = D^{n}$ . Every simple module is a quotient of $_{R} R = ⨁_{j} (column j) ≅ (D^{n})^{n}$ , so $D^{n}$ is the unique simple module up to isomorphism. An endomorphism commuting with all matrices is right multiplication by a scalar from $D$ , and composition reverses the order of scalar multiplication, giving $End_{M_{n} (D)} (D^{n}) ≅ D^{op}$ .

Advanced results Master

Theorem 1 (Schur's lemma). If $S$ and $T$ are simple $R$ -modules, every non-zero $R$ -homomorphism $S \to T$ is an isomorphism; in particular $End_{R} (S)$ is a division ring ^{[Lang Algebra Ch. XVII]}. This is the engine that converts the action of a ring on its simple modules into division-ring data, and it reappears as 07.01.02 (Schur's lemma) in the representation-theoretic register.

Theorem 2 (Jacobson density theorem). Let $S$ be a simple $R$ -module and $D = End_{R} (S)$ , so that $S$ is a right $D$ -vector space. Then the image of $R$ in $End_{D} (S)$ is dense: for every $D$ -linearly independent finite set $s_{1}, \dots, s_{n} \in S$ and every target tuple $t_{1}, \dots, t_{n} \in S$ , there exists $r \in R$ with $r s_{i} = t_{i}$ for all $i$ ^{[Jacobson 1945]}. If in addition $S$ is finite-dimensional over $D$ , the map $R \to End_{D} (S) ≅ M_{n} (D^{op})$ is surjective.

Theorem 3 (Artin-Wedderburn). A ring $R$ is semisimple if and only if $$ R ;\cong; \prod_{i=1}^{r} M_{n_i}(D_i) $$ for division rings $D_{i}$ and integers $n_{i} \geq 1$ ; the factors are determined up to permutation and isomorphism, $r$ equals the number of isomorphism classes of simple $R$ -modules, and $D_{i} ≅ End_{R} (S_{i})^{op}$ where $S_{i}$ is the $i$ -th simple module appearing $n_{i}$ times in $R$ ^{[Wedderburn 1908]}. The original division-algebra case is Wedderburn 1908; the Artinian-ring generality is Artin 1927 ^{[Artin 1927]}.

Theorem 4 (Radical as obstruction, Artinian case). If $R$ is left-Artinian then $J (R)$ is nilpotent, $R / J (R)$ is semisimple, and $R$ is semisimple if and only if $J (R) = 0$ . Thus $J (R)$ is the precise measurable obstruction to semisimplicity, and the structure theorem applies to the semisimple quotient $R / J (R)$ of any Artinian ring ^{[Lang Algebra Ch. XVII]}.

Theorem 5 (Maschke as the special case). If $G$ is a finite group and $k$ a field whose characteristic does not divide $∣ G ∣$ , then the group algebra $k [G]$ is semisimple, so $k [G] ≅ \prod_{i} M_{n_{i}} (D_{i})$ . For $k = C$ the division rings are all $C$ , giving $C [G] ≅ \prod_{i} M_{d_{i}} (C)$ with $\sum_{i} d_{i}^{2} = ∣ G ∣$ , where the $d_{i}$ are the dimensions of the irreducible representations ^{[Lang Algebra Ch. XVIII]}. This is 07.02.01 (Maschke's theorem) and 07.01.05 (group algebra and regular representation) seen as one instance of the general theorem.

Synthesis. The bridge of this unit is that three apparently separate facts — Schur's lemma, the density theorem, and Maschke's averaging — are one structure viewed from three sides. The foundational reason is that a simple module $S$ turns its ring into a dense subring of $D$ -linear maps, and the Artinian finiteness forces density to become surjectivity, which is exactly the matrix-ring presentation. This is exactly the upgrade that generalises the vector-space splitting "every subspace has a complement" into the module-theoretic semisimplicity condition, and it is dual to the radical picture: where Artin-Wedderburn describes the clean quotient, the Jacobson radical isolates the tangled kernel, and putting these together gives $R / J (R) ≅ \prod M_{n_{i}} (D_{i})$ for every Artinian ring. The central insight is that the representation theory of finite groups is not a separate subject but the special case $R = C [G]$ , so the character-degree identity $\sum d_{i}^{2} = ∣ G ∣$ is the same bookkeeping as $dim_{k} R = \sum n_{i}^{2} dim_{k} D_{i}$ that controls every semisimple ring.

Full proof set Master

Proposition 1 (Wedderburn's structure of a simple Artinian ring, via density). Let $R$ be a left-Artinian ring that is simple (no two-sided ideals other than $0$ and $R$ ) and semisimple. Then $R ≅ M_{n} (D)$ for a division ring $D$ and some $n \geq 1$ .

Proof. Since $R$ is semisimple, $_{R} R = ⨁_{j = 1}^{m} L_{j}$ is a finite direct sum of simple left ideals. All the $L_{j}$ are isomorphic: the set of left ideals isomorphic to a fixed $L_{1}$ spans a two-sided ideal (it is stable under right multiplication because right multiplication $L_{j} \to L_{j} r$ is an $R$ -module map onto an isomorphic image), and simplicity of $R$ forces this two-sided ideal to be all of $R$ . Write $S = L_{1}$ and $n = m$ , so $_{R} R ≅ S^{n}$ .

Let $D = End_{R} (S)$ , a division ring by Schur (Theorem 1). Then $$ R ;\cong; \operatorname{End}_R({}_RR) ;\cong; \operatorname{End}_R(S^n) ;\cong; M_n!\big(\operatorname{End}_R(S)\big) = M_n(D), $$ where the first isomorphism is $r \mapsto (x \mapsto x r)$ (right multiplication, an $R$ -endomorphism of $_{R} R$ , an anti-isomorphism that is absorbed into the opposite convention), and the third is the standard matrix description of endomorphisms of a direct sum of $n$ copies of one module. The Jacobson density theorem (Theorem 2) is what guarantees $R$ surjects onto the full endomorphism ring $End_{D} (S)$ when $S$ is finite-dimensional over $D$ , so the displayed map is onto and not merely injective. Hence $R ≅ M_{n} (D)$ . $□$

Proposition 2 (Artin-Wedderburn from the simple case). Every semisimple ring $R$ is a finite product $\prod_{i = 1}^{r} M_{n_{i}} (D_{i})$ of matrix rings over division rings.

Proof. Group the simple left ideals in $_{R} R = ⨁ L_{j}$ into isomorphism classes $S_{1}, \dots, S_{r}$ , with $S_{i}$ occurring $n_{i}$ times, so $_{R} R ≅ ⨁_{i = 1}^{r} S_{i}^{n_{i}}$ . Let $B_{i}$ be the sum of all left ideals isomorphic to $S_{i}$ ; this isotypic component is a two-sided ideal (right multiplication preserves isomorphism class as in Proposition 1). Distinct isotypic components annihilate each other, because a non-zero product would give a non-zero $R$ -map $S_{i} \to S_{i^{'}}$ between non-isomorphic simples, contradicting Schur. Therefore $R = ⨁_{i} B_{i}$ as a ring direct product, and each $B_{i}$ is a simple Artinian semisimple ring with a single isomorphism class of simple module. Proposition 1 gives $B_{i} ≅ M_{n_{i}} (D_{i})$ with $D_{i} = End_{R} (S_{i})^{op}$ , whence $R ≅ \prod_{i = 1}^{r} M_{n_{i}} (D_{i})$ . Uniqueness of the factors follows because $r$ is the number of isomorphism classes of simple modules and $(n_{i}, D_{i})$ are recovered as the multiplicity and opposite endomorphism ring of $S_{i}$ . $□$

Proposition 3 (Maschke's theorem). Let $G$ be a finite group and $k$ a field with $char k ∤ ∣ G ∣$ . Then $k [G]$ is semisimple.

Proof. It suffices to show every submodule $W$ of every $k [G]$ -module $V$ is a direct summand. Choose a $k$ -linear projection $π : V \to W$ (a complement exists as vector spaces). Average it over the group to obtain $$ \bar{\pi} ;=; \frac{1}{|G|} \sum_{g \in G} g ,\pi, g^{-1}, $$ which is well-defined because $∣ G ∣$ is invertible in $k$ . Each summand maps $V$ into $W$ (since $W$ is $G$ -stable), so $\overset{π}{ˉ} (V) \subseteq W$ ; and for $w \in W$ one has $π (g^{- 1} w) = g^{- 1} w$ , so $\overset{π}{ˉ} (w) = w$ . Thus $\overset{π}{ˉ}$ is a $k [G]$ -linear projection onto $W$ , and $V = W \oplus ker \overset{π}{ˉ}$ as $k [G]$ -modules. Every submodule is a direct summand, so $k [G]$ is semisimple. $□$

Connections Master

Schur's lemma 07.01.02. The representation-theoretic Schur's lemma — that intertwiners between irreducible representations are zero or isomorphisms, and self-intertwiners are scalars over an algebraically closed field — is precisely Theorem 1 applied to the simple $C [G]$ -modules. The division ring $End_{R} (S)$ collapses to $C$ exactly when the ground field is algebraically closed, which is why the complex character theory sees only matrix blocks over $C$ . This is the foundational reason the density-theorem proof here and the orthogonality relations there share an engine.
Maschke's theorem 07.02.01. Maschke's averaging is the special case of semisimplicity for $R = k [G]$ with $char k ∤ ∣ G ∣$ , established in Proposition 3. The group-averaged projection $\overset{π}{ˉ}$ is the concrete complement whose abstract existence is the content of the direct-summand criterion proved at Intermediate tier, so the rep-theory strand is the $C [G]$ shadow of the general Artin-Wedderburn structure theory.
Group algebra and the regular representation 07.01.05. The decomposition $C [G] ≅ \prod_{i} M_{d_{i}} (C)$ is exactly the Wedderburn decomposition of the regular representation: the left regular module $C [G]$ splits as $⨁_{i} S_{i}^{d_{i}}$ , recovering each irreducible $S_{i}$ with multiplicity equal to its dimension $d_{i}$ . The identity $\sum_{i} d_{i}^{2} = ∣ G ∣$ is the dimension count $dim_{C} C [G] = \sum_{i} d_{i}^{2}$ from the matrix-block decomposition, tying the abstract structure theorem to the concrete character table.
Jordan canonical form 01.01.11. The module-over- $k [t]$ viewpoint behind rational and Jordan canonical forms is the commutative cousin of this theory: $k [t] / (f)$ is semisimple exactly when $f$ is squarefree, and the radical $J (k [t] / (f))$ is generated by the repeated-factor part, mirroring how the Jacobson radical obstructs semisimplicity here. The primary decomposition of a single operator is the abelian instance of the isotypic decomposition $R = ⨁ B_{i}$ used in Proposition 2.

Historical & philosophical context Master

Wedderburn 1908 proved, in On hypercomplex numbers ^{[Wedderburn 1908]}, that every finite-dimensional simple algebra over a field is a matrix algebra over a division algebra, completing a programme begun by Benjamin Peirce and Cartan on the classification of "hypercomplex number systems." His method ran through nilpotent ideals and idempotent decomposition, isolating the radical as the obstruction to the clean structure. The result reorganised what had been a zoo of low-dimensional algebra tables into a single classification governed by division rings.

Artin 1927 ^{[Artin 1927]} recast the theorem in terms of the descending chain condition, freeing it from finite dimensionality over a field and producing the form now called Artin-Wedderburn: the structure theory holds for any semisimple ring, with the Artinian hypothesis the right finiteness condition. Jacobson 1945 ^{[Jacobson 1945]} then removed even the chain condition from the core mechanism by proving the density theorem for primitive rings, showing that a ring acting faithfully and irreducibly is a dense ring of linear transformations of a vector space over a division ring. The streamlined proof of Artin-Wedderburn via density, in the form used in Propositions 1 and 2, is due to Rieffel 1965 ^{[Rieffel 1965]}, who observed that the simple ring is recovered as its own endomorphism ring acting on a minimal left ideal. Philosophically the arc is a passage from concrete computation with multiplication tables to a structural principle: indivisible representations are the atoms, and the ring is reconstructed from how it acts on them.

Bibliography Master

@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{Artin1927,
  author = {Artin, Emil},
  title = {Zur Theorie der hyperkomplexen Zahlen},
  journal = {Abhandlungen aus dem Mathematischen Seminar der Universit\"at Hamburg},
  volume = {5},
  year = {1927},
  pages = {251--260},
}

@article{Jacobson1945,
  author = {Jacobson, Nathan},
  title = {Structure theory of simple rings without finiteness assumptions},
  journal = {Transactions of the American Mathematical Society},
  volume = {57},
  year = {1945},
  pages = {228--245},
}

@article{Rieffel1965,
  author = {Rieffel, Marc A.},
  title = {A general Wedderburn theorem},
  journal = {Proceedings of the National Academy of Sciences},
  volume = {54},
  year = {1965},
  pages = {1513},
}

@book{Lang2002,
  author = {Lang, Serge},
  title = {Algebra},
  edition = {3rd revised},
  publisher = {Springer},
  year = {2002},
  series = {Graduate Texts in Mathematics 211},
}

@book{Lam2001,
  author = {Lam, Tsit-Yuen},
  title = {A First Course in Noncommutative Rings},
  edition = {2nd},
  publisher = {Springer},
  year = {2001},
  series = {Graduate Texts in Mathematics 131},
}

Prerequisites

01.02.06
01.02.10
01.01.11
07.01.02

Tier anchors

beginner: Lang Algebra Ch. XVII §1 (semisimple modules, informal); Dummit-Foote §18.1 worked C[G] decompositions
intermediate: Lang Algebra Ch. XVII §§3-5; Dummit-Foote §18.2 (Artin-Wedderburn); Lam A First Course in Noncommutative Rings §3 (density theorem)
master: Wedderburn 1908 On hypercomplex numbers (Proc. LMS); Artin 1927 Zur Theorie der hyperkomplexen Zahlen; Jacobson 1945 Structure theory of simple rings; Rieffel 1965 density-theorem proof

References

Wedderburn — On hypercomplex numbers · 1908, Proceedings of the London Mathematical Society (2) 6, 77-118
Artin — Zur Theorie der hyperkomplexen Zahlen · 1927, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 5, 251-260
Jacobson — Structure theory of simple rings without finiteness assumptions · 1945, Transactions of the American Mathematical Society 57, 228-245
Rieffel — A general Wedderburn theorem · 1965, Proceedings of the National Academy of Sciences 54, 1513
Lang — Algebra · Ch. XVII
Lam — A First Course in Noncommutative Rings · GTM 131, §§2-4

Estimated time

beginner: 18m
intermediate: 45m
master: 85m