Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem

Given: The hypotheses and objects in the Statement.

For every module homomorphism $f:M\to N$, there is a module isomorphism $M\mathrm{/}\ker f\cong \mathrm{im}f,$ given by $m+\ker f\mapsto f(m)$. (First isomorphism theorem for modules: $M\mathrm{/}\ker f\cong \mathrm{im}f$).

For submodules $A,B\le M$, the intersection $A\cap B$ and the sum $A+B:=\{a+b:a\in A,b\in B\}$ are submodules of $M$. (The one-step submodule criterion; intersections and sums of submodules are submodules).

For $N\le M$, the additive cosets $m+N$ form the quotient module $M\mathrm{/}N$ under the well-defined scalar action $r(m+N):=rm+N\mathrm{.}$ (Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets).

Given: The hypotheses and objects in the Statement.

For every module homomorphism $f:M\to N$, there is a module isomorphism $M\mathrm{/}\ker f\cong \mathrm{im}f,$ given by $m+\ker f\mapsto f(m)$. (First isomorphism theorem for modules: $M\mathrm{/}\ker f\cong \mathrm{im}f$).

For $N\le M$, the additive cosets $m+N$ form the quotient module $M\mathrm{/}N$ under the well-defined scalar action $r(m+N):=rm+N\mathrm{.}$ (Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets).

Let $f:M\to P$ be a module homomorphism and let $N\le M$ satisfy $N\subseteq \ker f$. There is a unique module homomorphism $\bar{f}:M\mathrm{/}N⟶P$ such that $\bar{f}(m+N)=f(m)$, equivalently $f=\bar{f}\circ \pi$. (A module homomorphism vanishing on $N$ factors uniquely through $M\mathrm{/}N$).

Given: The hypotheses and objects in the Statement.

For $N\le M$, the additive cosets $m+N$ form the quotient module $M\mathrm{/}N$ under the well-defined scalar action $r(m+N):=rm+N\mathrm{.}$ (Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets).

Let $f:M\to P$ be a module homomorphism and let $N\le M$ satisfy $N\subseteq \ker f$. There is a unique module homomorphism $\bar{f}:M\mathrm{/}N⟶P$ such that $\bar{f}(m+N)=f(m)$, equivalently $f=\bar{f}\circ \pi$. (A module homomorphism vanishing on $N$ factors uniquely through $M\mathrm{/}N$).

If $N\le L\le M$, then $L\mathrm{/}N$ is a submodule of $M\mathrm{/}N$ and $(M\mathrm{/}N)\mathrm{/}(L\mathrm{/}N)\cong M\mathrm{/}L\mathrm{.}$ (Third isomorphism theorem for modules).

Given: The hypotheses and objects in the Statement. The adopted axiom of dependent choice is assumed for the one direction identified in the Statement; it is not cited as a forward dependency.

A left $R$-module $M$ is Noetherian when every submodule of $M$ is finitely generated (def-generated-cyclic-finitely-generated-and-free-modules). This finite-generation definition is the convention; its equivalence with ACC and the maximal condition is proved in thm-equivalent-characterizations-of-noetherian-modules. (Noetherian modules: every submodule is finitely generated).

Given: The hypotheses and objects in the Statement. The adopted axiom of dependent choice is assumed for the one direction identified in the Statement; it is not cited as a forward dependency.

A left $R$-module $M$ is Artinian when every descending chain $M_0\supseteq M_1\supseteq \cdots$ of submodules stabilizes: there is $N$ such that $M_n=M_N$ for all $n\ge N$. This is the descending chain condition. (Artinian modules by the descending chain condition).

Given: The hypotheses and objects in the Statement.

A left $R$-module $M$ is Noetherian when every submodule of $M$ is finitely generated (def-generated-cyclic-finitely-generated-and-free-modules). This finite-generation definition is the convention; its equivalence with ACC and the maximal condition is proved in thm-equivalent-characterizations-of-noetherian-modules. (Noetherian modules: every submodule is finitely generated).

A left $R$-module $M$ is Artinian when every descending chain $M_0\supseteq M_1\supseteq \cdots$ of submodules stabilizes: there is $N$ such that $M_n=M_N$ for all $n\ge N$. This is the descending chain condition. (Artinian modules by the descending chain condition).

For submodules $L,N\le M$, there is a canonical isomorphism $L\mathrm{/}(L\cap N)\cong (L+N)\mathrm{/}N\mathrm{.}$. (Second isomorphism theorem for modules).

For $N\le M$, inverse image and quotient induce mutually inverse inclusion-preserving bijections between submodules of $M\mathrm{/}N$ and submodules of $M$ containing $N$. They preserve sums, intersections, and successive quotients. (Correspondence theorem for submodules of a quotient module).

A short exact sequence is an exact sequence $0⟶N\stackrel{i}{\to }M\stackrel{p}{\to }Q⟶0;$ thus $i$ is injective, $p$ is surjective, and $\mathrm{im}i=\ker p$. (Exact sequences and short exact sequences of modules).

A module is Noetherian if and only if every ascending chain of submodules stabilizes. (Finite generation, ACC, and maximal-condition characterizations of Noetherian modules).

A module is Artinian if and only if every descending chain of submodules stabilizes. (DCC and minimal-condition characterizations of Artinian modules).

Given: The hypotheses and objects in the Statement.

In a short exact sequence $0\to N\to M\to Q\to 0$, the module $M$ is Noetherian if and only if $N$ and $Q$ are Noetherian; the same equivalence holds with “Artinian” in place of “Noetherian”. (Noetherian and Artinian conditions are each exact in short exact sequences).

Let $R$ be a unital ring and $(M_i{)}_{i\in I}$ a family of left $R$-modules (def-left-and-right-modules). Their direct product is the module ${\prod }_{i\in I}{M}_i$ with coordinatewise operations. The support of $m=(m_i)$ is $\{i\in I:m_i\mathrm{\ne }0\}$, and the direct sum is the submodule ${⨁}_{i\in I}{M}_i=\{m\in {\prod }_{i\in I}{M}_i:\mathrm{supp}(m)\text{ is finite}\}$ (def-submodule). (The direct sum of an indexed family of modules).

Given: The hypotheses and objects in the Statement.

A unital ring $R$ is left Noetherian when its left regular module ${}_{R}R$ is Noetherian, and right Noetherian when the right regular module $R_R$ is Noetherian. Unqualified “Noetherian ring” means left Noetherian here; the side is stated whenever both notions occur. (Left and right Noetherian rings).

Let $M$ be a left $R$-module and $S\subseteq M$. The submodule generated by $S$ is $⟨S{⟩}_R:=\bigcap \{N\le M:S\subseteq N\}\mathrm{.}$ The family is nonempty because $M\le M$, and its intersection is a submodule by lem-submodule-criterion-sums-and-intersections. Thus $⟨S{⟩}_R$ is the smallest submodule of $M$ containing $S$, just as def-generated-subgroup defines a generated subgroup. (Generated submodule, cyclic and finitely generated modules, module basis and free module).

A finite direct sum is Noetherian if and only if every summand is Noetherian, and it is Artinian if and only if every summand is Artinian. The empty direct sum is included. (Finite direct sums preserve and reflect Noetherian and Artinian conditions).

For every left $R$-module $M$, the free module $R^{(M)}$ on its underlying set admits a canonical surjection ${\epsilon }_M:R^{(M)}\to M$, determined by ${\epsilon }_M(e_m)=m$. Consequently $M\cong R^{(M)}\mathrm{/}\ker {\epsilon }_M$. (Every module is a quotient of a free module).

In a short exact sequence $0\to N\to M\to Q\to 0$, the module $M$ is Noetherian if and only if $N$ and $Q$ are Noetherian. (Noetherian and Artinian conditions are each exact in short exact sequences).

Given: The hypotheses and objects in the Statement.

For a left $R$-module $M$, the following are equivalent: every submodule is finitely generated; every ascending chain of submodules stabilizes; and every nonempty family of submodules has a maximal member. The implication from ACC to the maximal condition uses dependent choice; the other displayed implications are choice-free. (Finite generation, ACC, and maximal-condition characterizations of Noetherian modules).

For a left $R$-module $M$, define ${\mathrm{End}}_R(M):={\mathrm{Hom}}_R(M,M)\mathrm{.}$ Addition is pointwise and multiplication is composition, $(fg)(m):=f(g(m))$. The ring laws and the identity endomorphism are established in prop-endomorphisms-form-a-ring. (The endomorphism ring ${\mathrm{End}}_R(M)$ under addition and composition).

Given: The hypotheses and objects in the Statement.

A composition series of a left $R$-module $M$ is a finite chain $0=M_0<M_1<\cdots <M_n=M$ whose factors $M_i\mathrm{/}{M}_{i-1}$ are simple. If such a series exists, the length ${\mathrm{\ell }}_R(M)$ is its number $n$ of factors; thm-jordan-holder-theorem-for-modules proves independence of the chosen series. The zero module has the empty series and length $0$. (Composition series and length of a module).

For a left $R$-module $M$, the following are equivalent: every submodule is finitely generated; every ascending chain of submodules stabilizes; and every nonempty family of submodules has a maximal member. The implication from ACC to the maximal condition uses dependent choice; the other displayed implications are choice-free. (Finite generation, ACC, and maximal-condition characterizations of Noetherian modules).

For a left $R$-module $M$, DCC is equivalent to the condition that every nonempty family of submodules has a minimal member. The implication from DCC to the minimal condition uses dependent choice. (DCC and minimal-condition characterizations of Artinian modules).

In a short exact sequence $0\to N\to M\to Q\to 0$, the module $M$ is Noetherian if and only if $N$ and $Q$ are Noetherian; the same equivalence holds with “Artinian” in place of “Noetherian”. (Noetherian and Artinian conditions are each exact in short exact sequences).

Given: The hypotheses and objects in the Statement.

A composition series of a left $R$-module $M$ is a finite chain $0=M_0<M_1<\cdots <M_n=M$ whose factors $M_i\mathrm{/}{M}_{i-1}$ are simple. If such a series exists, the length ${\mathrm{\ell }}_R(M)$ is its number $n$ of factors; thm-jordan-holder-theorem-for-modules proves independence of the chosen series. The zero module has the empty series and length $0$. (Composition series and length of a module).

For submodules $L,N\le M$, there is a canonical isomorphism $L\mathrm{/}(L\cap N)\cong (L+N)\mathrm{/}N\mathrm{.}$. (Second isomorphism theorem for modules).

If $N\le L\le M$, then $L\mathrm{/}N$ is a submodule of $M\mathrm{/}N$ and $(M\mathrm{/}N)\mathrm{/}(L\mathrm{/}N)\cong M\mathrm{/}L\mathrm{.}$. (Third isomorphism theorem for modules).

For $N\le M$, inverse image and quotient induce mutually inverse inclusion-preserving bijections between submodules of $M\mathrm{/}N$ and submodules of $M$ containing $N$. They preserve sums, intersections, and successive quotients. (Correspondence theorem for submodules of a quotient module).

Given: The hypotheses and objects in the Statement.

Any two composition series of a module have the same length, and their simple factors agree up to permutation and isomorphism. (Jordan–Hölder theorem for modules).

A composition series of a left $R$-module $M$ is a finite chain $0=M_0<M_1<\cdots <M_n=M$ whose factors $M_i\mathrm{/}{M}_{i-1}$ are simple. If such a series exists, the length ${\mathrm{\ell }}_R(M)$ is its number $n$ of factors; thm-jordan-holder-theorem-for-modules proves independence of the chosen series. The zero module has the empty series and length $0$. (Composition series and length of a module).

For $N\le M$, inverse image and quotient induce mutually inverse inclusion-preserving bijections between submodules of $M\mathrm{/}N$ and submodules of $M$ containing $N$. They preserve sums, intersections, and successive quotients. (Correspondence theorem for submodules of a quotient module).

Given: The hypotheses and objects in the Statement.

For a left $R$-module $M$, define ${\mathrm{End}}_R(M):={\mathrm{Hom}}_R(M,M)\mathrm{.}$ Addition is pointwise and multiplication is composition, $(fg)(m):=f(g(m))$. The ring laws and the identity endomorphism are established in prop-endomorphisms-form-a-ring. (The endomorphism ring ${\mathrm{End}}_R(M)$ under addition and composition).

For left $R$-modules $M,N$, the set ${\mathrm{Hom}}_R(M,N)$ of module homomorphisms is an abelian group under pointwise addition, with zero the zero homomorphism and inverse $(-f)(m)=-f(m)$ (def-module-homomorphism-kernel-image-and-cokernel, def-group). (The abelian group ${\mathrm{Hom}}_R(M,N)$ and maps induced by pre- and postcomposition).

For left $R$-modules $M,N$, a function $f:M\to N$ is an $R$-module homomorphism if $f(m+m^{\mathrm{\prime }})=f(m)+f(m^{\mathrm{\prime }})$ and $f(rm)=rf(m)$ for all $m,m^{\mathrm{\prime }}\in M$ and $r\in R$ (Module homomorphism and isomorphism, kernel, image and cokernel).

Given: The hypotheses and objects in the Statement.

For a unital ring $R$, the opposite ring $R^{\mathrm{o}\mathrm{p}}$ has the same underlying abelian group, identity, and addition as $R$, with multiplication $a\star b:=ba$. Associativity and both distributive laws follow from those of $R$ with the order reversed, and the same element $1$ is a two-sided identity. Thus the displayed operations really form a unital ring, including when $R$ is the zero ring. (The opposite ring $R^{\mathrm{o}\mathrm{p}}$).

For every left $R$-module $M$, pointwise addition and composition make ${\mathrm{End}}_R(M)$ a unital ring with identity ${\mathrm{id}}_M$. (Module endomorphisms form a ring under pointwise addition and composition).

Let $R$ be a ring. A left $R$-module is an abelian group $(M,+,0_M)$ with a scalar action $R\times M\to M$, $(r,m)\mapsto rm$, satisfying $r(m+n)=rm+rn,(r+s)m=rm+sm,(rs)m=r(sm),1_{R}m=m\mathrm{.}$ A right $R$-module has an action $M\times R\to M$, $(m,r)\mapsto mr$, with the analogous right-handed axioms. Unless “right” is stated, module means a unital left module. (Unital left and right modules over a ring; unqualified module means left module).

Given: The hypotheses and objects in the Statement.

For a left $R$-module $M$, define ${\mathrm{End}}_R(M):={\mathrm{Hom}}_R(M,M)\mathrm{.}$ Addition is pointwise and multiplication is composition, $(fg)(m):=f(g(m))$. The ring laws and the identity endomorphism are established in prop-endomorphisms-form-a-ring. (The endomorphism ring ${\mathrm{End}}_R(M)$ under addition and composition).

For left $R$-modules $M,N$, the set ${\mathrm{Hom}}_R(M,N)$ of module homomorphisms is an abelian group under pointwise addition, with zero the zero homomorphism and inverse $(-f)(m)=-f(m)$ (def-module-homomorphism-kernel-image-and-cokernel, def-group). (The abelian group ${\mathrm{Hom}}_R(M,N)$ and maps induced by pre- and postcomposition).

Let $(M_i{)}_{i\in I}$ be left $R$-modules and $N$ a left $R$-module. For every family of homomorphisms $f_i:M_i\to N$, there is a unique homomorphism $f:{⨁}_{i\in I}{M}_i⟶N$ such that $f\circ {\mathrm{ȷ}}_i=f_i$ for every $i$. It is given by $f((m_i))={\sum }_{i\in \mathrm{supp}(m)}{f}_i(m_i)\mathrm{.}$ For $I=\mathrm{\varnothing }$, this is the unique map $0\to N$. (Universal property of a direct sum of modules).

Given: The hypotheses and objects in the Statement.

Let $M$ be a left $R$-module and $S\subseteq M$. The submodule generated by $S$ is $⟨S{⟩}_R:=\bigcap \{N\le M:S\subseteq N\}\mathrm{.}$ The family is nonempty because $M\le M$, and its intersection is a submodule by lem-submodule-criterion-sums-and-intersections. Thus $⟨S{⟩}_R$ is the smallest submodule of $M$ containing $S$, just as def-generated-subgroup defines a generated subgroup. (Generated submodule, cyclic and finitely generated modules, module basis and free module).

Let $M$ be a left $R$-module. A subset $N\subseteq M$ is a submodule when it is a subgroup of the additive group of $M$ and is closed under scalars: $r\in R,n\in N⟹rn\in N\mathrm{.}$ The operations on $N$ are the restrictions of those of $M$. Write $N\le M$ when the ring and module are understood. (Submodule of a module).

Assume the Axiom of Choice (def-axiom-of-choice). Let $(P,\le )$ be a nonempty poset in which every chain has an upper bound. Then $P$ has a maximal element (def-maximal-element). Note the hypothesis asks only for an upper bound, not a least upper bound, and the conclusion asserts only that a maximal element exists, never that a greatest one does. (Zorn's lemma).

Given: The hypotheses and objects in the Statement.

A left $R$-module is semisimple when it is an internal direct sum of simple submodules, allowing the empty direct sum. Hence the zero module is semisimple. (Semisimple modules as direct sums of simple modules).

Assuming the Axiom of Choice, every finitely generated nonzero module has a maximal proper submodule. (Under Choice, every finitely generated nonzero module has a maximal proper submodule).

Assume the Axiom of Choice (def-axiom-of-choice). Let $(P,\le )$ be a nonempty poset in which every chain has an upper bound. Then $P$ has a maximal element (def-maximal-element). Note the hypothesis asks only for an upper bound, not a least upper bound, and the conclusion asserts only that a maximal element exists, never that a greatest one does. (Zorn's lemma).

Let $(M_i{)}_{i\in I}$ be left $R$-modules and $N$ a left $R$-module. For every family of homomorphisms $f_i:M_i\to N$, there is a unique homomorphism $f:{⨁}_{i\in I}{M}_i⟶N$ such that $f\circ {\mathrm{ȷ}}_i=f_i$ for every $i$. It is given by $f((m_i))={\sum }_{i\in \mathrm{supp}(m)}{f}_i(m_i)\mathrm{.}$ For $I=\mathrm{\varnothing }$, this is the unique map $0\to N$. (Universal property of a direct sum of modules).

The socle $\mathrm{Soc}(M)$ is the sum of all simple submodules of $M$. (The socle as the sum of all simple submodules).