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Noetherian and Artinian

• Every submodule of $M$ is finitely generated.

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If the $B$-module $B$ satisfies condition in ^4af1b0, then we say $B$ is a Noetherian ring.

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Lemma

For an $R$-module $M$, TFAE:

• Every decreasing sequence of submodules is stationary.
• Every non-empty family of submodules has a minimal elements.

Say the ring $A$ is a Noetherian ring (rep. Artinian ring) if $A$ is a Noetherian (rep. Artinian) $A$-module.

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Remark. If $M$ is a Noetherian (or Artinian) module, we also say that $M$ satisfies the maximal (or minimal) condition on submodules.

• $M$ is Artinian iff $M^{\prime }$ and $M^{\prime \prime }$ are Artinian.

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If $M_1,\cdots ,M_n$ are Noetherian (Artinian) $A$-module, then so is ${\oplus }_{i=1}^n{M}_i$.

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If $A$ is Noetherian, then any finitely generated $A$-module is a Noetherian $A$-module.

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Composition Series of Modules

Say its length is $n$. Say a chain is a composition series if no extra submodule can be inserted, that is, $M_{i-1}/M_i$ is a simple module for all $1⩽i⩽n$.

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$M$ has a finite length composition series iff $M$ is Noetherian and Artinian.

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Theorem

Fact (Jordan-Holder theorem for modules of finite length). if ${\left(M_i\right)}_{0<1<\pi }$ and ${\left(M_i^{\prime }\right)}_{0<1<n}$ are any two composition series of $M$, there is a one-to-one correspondence between the set of quotients ${\left(M_{i-1}/M_i\right)}_{1<1<n}$ and the set of quotients ${\left(M_{i-1}^{\prime }/M_i^{\prime }\right)}_{1<1<n}$, such that corresponding quotients are isomorphic. The proof is the same as for finite groups.

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Schur Lemma and Socle

Schur lemma

Let $S_1$ and $S_2$ be simple $R$-modules. Then ${\mathrm{Hom}}_R(S_1,S_2)=0$ unless $S_1\simeq S_2$ in which case the endomorphism ring ${\mathrm{End}}_R(S_1)$ is a division ring.

Lemma

Let $R$ be a ring with identity $1$. Suppose that $U=S_1+\cdots +S_n$ is an $R$-module that can be expressed as the sum of finitely many simple submodule. If $V$ is any submodule of $U$, then there is a subset $I\subseteq \{1,\cdots ,n\}$ such that $U=V\oplus ({\oplus }_{i\in I}{S}_i)$.

In particular, $V$ is a direct summand of $U$, and $U$ is semisimple.

Theorem

Let $U$ be an $R$-module. TFAE:

• $U$ is semisimple;
• $U$ is a sum of simple $R$-modules;
• any $R$-submodule of $U$ is a direct summand of $U$.

\begin{proof} See here. \end{proof}

Corollary

Let $U$ be the $R$-module with composition series of finite composition length.

• The sum of all the simple submodule of $U$ is a semisimple module that is the unique largest semisimple submodule of $U$.
• The sum of all submodules of $U$ isomorphic to some given simple module $S$ is a submodule isomorphic to a direct sum of copies of $S$. It is the unique largest submodule with this property.

Definition

The unique largest semisimple submodule of $U$ is called the socle of $U$, denote $\mathrm{soc}(U)$.

Corollary

Let $U=S_1^{a_1}\oplus \cdots \oplus S_r^{a_r}$ be a semisimple submodule where the $S_i$ are non-isomorphic simple $R$-modules. Then each submodule $S_i^{a_i}$ is uniquely determined and characterized as unique largest submodule of $U$ expressed as a direct sum of copies of $S_i$.

Theorem

Let $R$ be a ring with $1$. Let ${}_{R}V=V_1\oplus V_2\oplus \cdots \oplus V_n$ be a direct sum of simple $R$-modules $V_i$ such that $V_i\simeq V_j$ for any $i,j$. Then ${\mathrm{End}}_R(V)$ is isomorphic to the full matrix ring of degree $n$ over the division ring ${\mathrm{End}}_R(V_1)$.

Projective Cover

Definition

An epimorphism of modules $f:U\twoheadrightarrow V$ is called essential if no proper submodule of $U$ is mapped surjectively onto $V$ by $f$. Equivalently, whenever $g:W\to U$ is a map such that $fg$ is an epimorphism, then $g$ is an epimorphism.

Definition

A projective cover of a module $U$ is an essential epimorphism $P\stackrel{\tau }{\twoheadrightarrow }U$ where $P$ is projective.

Remark. Projective cover does not always exist.

Proposition

Suppose that $f:P\to U$ is a projective cover of a module $U$ and $g:Q\to U$ is an epimorphism where $Q$ is projective. Then $Q=Q_1\oplus Q_2$ so that $g$ has the component $g=(g_1,0)$ with respect to the direct sum decomposition and $g_1:Q_1\to U$ satisfies that the diagram commutes

where $\gamma$ is an isomorphism.

If any exists, the projective covers of a module $U$ are all isomorphic, by isomorphisms that commute with the essential epimorphism.

\begin{proof} See ^9so7ao. \end{proof}