4 250318

Radical of Modules

maximal submodule

A maximal submodule $N$ of $M$ is a proper submodule $N$ satisfying $N\subseteq U\subseteq M$ implies $U=N$ or $U=M$. Equivalently, $M/N$ is simple.

Definition

Define $\mathrm{Rad}(M)={\cap }_N\{\text{mbox}maximalsubmodulesN\subseteq M\}$. If $M$ does not have maximal submodules, then define $\mathrm{Rad}(M)=M$.

Lemma

Let $W$ be a proper $R$-submodule of ${}_{R}V$. If $V/W$ is finitely generated over $R$, then there exists a maximal $R$-submodule of $V$ containing $W$. In particular, if $V$ is finitely generated, then there exists a maximal submodule of $V$ containing $W$.

\begin{proof} See ^fbacdf. \end{proof}

Corollary

For every proper left ideal $I$ of $R$, there exists a maximal left ideal of $R$ containing $I$.

Lemma

Let $M,N$ be finitely generated $R$-modules, and let $f:M\to N$ be $R$-morphism.

• $f(\mathrm{Rad}(M))\subseteq \mathrm{Rad}(N)$.
• If $f$ is an epimorphism with $\ker f\subseteq \mathrm{Rad}(M)$, then $\mathrm{Rad}(N)=f(\mathrm{Rad}(M))$.

\begin{proof} See ^27mw0t. \end{proof}

Lemma

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

• Suppose that $M_1,\cdots ,M_n$ are maximal submodules of $U$. Then there is a subset $I\subseteq \{1,\cdots ,n\}$ such that

  $$U/(M_1\cap \cdots \cap M_n)\simeq {\oplus }_{i\in I}U/M_i$$

• Suppose that $U$ is Artinian, then $U/\mathrm{Rad}U$ is a semisimple and $\mathrm{Rad}U$ is the unique smallest submodule of $U$ with semisimple quotient.

\begin{proof} See ^r8oe32. \end{proof}

Nakayama lemma

Nakayama's lemma

Let $U$ be Noetherian(or finitely generated). Then $U\to U/\mathrm{Rad}U$ is essential. Equivalently, if $V$ is a submodule of $U$ with $V+\mathrm{Rad}U=U$, then $V=U$.

\begin{proof} See ^3f1b99. \end{proof}

Proposition

• Suppose that $f:U\to V$ and $g:V\to W$ are $R$-morphisms of $R$-modules. If two of $f,g,gf$ are essential, then so is the third.
• Let $f:U\to V$ be a homomorphism of Noetherian modules. Then $f$ is an essential epimorphism iff the radical quotient $U/\mathrm{Rad}U\to V/\mathrm{Rad}V$ is an essential epimorphism. In fact, if it holds, then the radical quotient is an isomorphism.
• Let $f_i:U_i\to V_i$ be homomorphisms of Noetherian modules for $i=1,\cdots ,n$, then $f_i$ are all essential epimorphisms iff $\oplus f_i:\oplus U_i\to \oplus V_i$ is an essential epimorphism.

\begin{proof} See ^q7sonb. \end{proof}

Corollary

If $P$ and $Q$ are projective Noetherian modules over a ring, then $P\simeq Q$ iff $P/\mathrm{Rad}P\simeq Q/\mathrm{Rad}Q$.

\begin{proof} See ^9ry2ru. \end{proof}

Semisimple

Definition

A ring $R$ with identity is called semisimple if ${}_{R}R$ is semisimple as a left $R$-modules.

Remark. ${}_{R}R$ is called (left) regular module.

Theorem

TFAE for an Artinian ring $R$.

• $R$ is a semisimple Artinian ring.
• Every $R$-module $V$ is semisimple.

\begin{proof} See ^7dokb6. \end{proof}

Some Isomorphisms

Lemma

Let $M$ be a left $R$-module. Then $M\simeq {\mathrm{Hom}}_R({}_R{R}_R,{}_{R}M)$.

\begin{proof} See ^cj265g \end{proof}

Corollary

For any ring $R$ with identity, one has ${\mathrm{End}}_R({}_{R}R)\simeq R^{\mathrm{op}}$ as rings.

\begin{proof} See ^5830df \end{proof}

Lemma

Let $R$ be a ring, and let $M_n(R)$ be the set of all square matrices of degree $n$ over $R$. Then ${\mathrm{M}}_n(R{)}^{\mathrm{op}}\simeq {\mathrm{M}}_n(R^{\mathrm{op}})$.

\begin{proof} See ^hvg28s. \end{proof}

Artin–Wedderburn

Let $A$ be a semisimple Artinian ring with identity. Then $A$ is a direct sum of matrix rings over division rings. Specifically, if ${}_{A}A\simeq S_1^{n_1}\oplus \cdots \oplus S_r^{n_r}$ where $S_1,\cdots ,S_r$ are non-isomorphic simple modules occurring with multiplications $n_1,\cdots ,n_r$, then $A\simeq {\mathrm{M}}_{n_1}(D_1)\oplus \cdots \oplus {\mathrm{M}}_{n_r}(D_r)$ where $D_i={\mathrm{End}}_A(S_i{)}^{\mathrm{op}}$.

\begin{proof} See ^f43ojz. \end{proof}

Simple Modules

Definition

A ring is called simple if it has no ideal other than $0$ and $R$.

Example. ${\mathrm{M}}_n(D)$ is a simple ring if $D$ is a division ring.

Notice that

$${\mathrm{M}}_n(D)=\left\{\left(\begin{array}{cccc}\ast & 0& \cdots & 0\\ \ast & 0& \cdots & 0\\ ⋮& ⋮& & ⋮\\ \ast & 0& \cdots & 0\end{array}\right)\right\}\oplus \cdots \oplus \left\{\left(\begin{array}{cccc}0& \cdots & 0& \ast \\ 0& \cdots & 0& \ast \\ ⋮& & ⋮& ⋮\\ 0& \cdots & 0& \ast \end{array}\right)\right\}$$

and each part is a simple left ${\mathrm{M}}_n(D)$-module.

Algebras

Definition

Let $R$ be a commutative ring with identity, and let $A$ be a ring with identity. We call $A$ an $R$-algebra if $A$ is an $R$-module such that $a(xy)=(ax)y=x(ay)$ for any $a\in R$ and any $x,y\in A$. Equivalently, there is a ring homomorphism $R\to A$ such that the image of $R$ lies in the center of $A$.

Moreover, if $R$ is a field and $A$ is a finite-dimensional over $R$, then we shall say that the algebra is finite-dimensional, which is both Noetherian and Artinian.

Examples.

• Suppose $R$ is a commutative ring with identity and $G$ is a finite group. Define group algebra/group ring as $RG=\{{\sum }_{g\in G}{a}_{g}g:a_g\in R\}$, which is a free $R$-module with basis $\{g:g\in G\}$, then $RG$ is a $R$-algebra.
• For a field $F$, the polynomial ring $F[x]$ is a $F$-algebra.
• For a commutative ring $R$, ${\mathrm{M}}_n(R)$ is a $R$-algebra, where $r\mapsto \mathrm{diag}(r,\cdots ,r)$.

Schur

Let $A$ be a $k$-algebra, where $k$ is a field. Let $S$ be a simple $A$-module. Then ${\mathrm{End}}_A(S)$ is a division ring, which is a $k$-algebra.

If $A$ is finite dimensional over $k$, so is ${\mathrm{End}}_A(S)$.

Furthermore, if $A$ is finite dimensional and $k$ is algebraically closed, then ${\mathrm{End}}_A(S)\simeq k$.

\begin{proof} See ^evw7zf. \end{proof}