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Annihilator

Definition

Let $R$ be a ring, and let $M$ be a left $R$-module. For any $x\in M$, define

$${\mathrm{Ann}}_R(x)=\{r\in R:rx=0\}$$

as the annihilator of $x$ in $R$. Notice that ${\mathrm{Ann}}_R(x)$ is a left ideal of $R$.

For a set $X\subseteq M$, define ${\mathrm{Ann}}_R(X)={\cap }_{x\in X}{\mathrm{Ann}}_{R}x$. If $N$ is a $R$-submodule of $M$, then ${\mathrm{Ann}}_R(N)$ is a $2$-sided ideal of $R$.

Lemma

Let $R$ be a ring with identity $1$. Let $S$ be a left simple $R$-module. Then $S$ is a quotient of the left regular module ${}_{R}R$.

\begin{proof} See ^09efb3. \end{proof}

Theorem

A finite-dimensional semisimple $k$-algebra $A$ over a field $k$ with $k=\overline{k}$ is isomorphism to a direct sum of matrix algebras over $k$.

\begin{proof} See ^88dlwq. \end{proof}

Corollary

If $A$ is a finite-dimensional $k$-algebra, then any simple $A$-module is of finite-dimensional as $k$-vector space.

Proposition

Let $A$ be a finite-dimensional semisimple $k$-algebra over a field $k$. In any decomposition ${}_{A}A=S_1^{n_1}\oplus \cdots \oplus S_r^{n_r}$ where the $S_i$ are pairwise non-isomorphic simple $A$-modules, we have $S_1,\cdots ,S_r$ is a complete set of representative of the isomorphism of simple $A$-modules.

If moreover $k=\overline{k}$, then $n_i={\dim }_k{S}_i$ and $\dim {}_{A}A={\sum }_{i=1}^r{n}_i^2$.

\begin{proof} See ^yo1zc8. \end{proof}

Idempotents

Idempotents（幂等元）

Let $R$ be a ring with identity. An idempotent in $R$ is a non-zero element $e$ with $e^2=e$.

Fact.

• $eRe$ is a ring with identity $e$, where $eRe=\{eae:a\in R\}$.
• For any $x\in Re$, $x=re$ for some $r\in R$ and then $xe=ree=re=x$.

Definition

Two idenmpotents $e_1,e_2$ are called orthogonal if $e_1{e}_2=0=e_2{e}_1$. More generally, $n$ idempotents $e_1,\cdots ,e_n$ are called orthogonal if $e_i{e}_j=0=e_j{e}_i$.

Fact. If $e_1,\cdots ,e_n$ are orthogonal idempotent, then $e_1+\cdots +e_n$ is also an idempotent.

Definition

• An idempotent $e$ is called primitive if it is not a sum of two orthogonal idempotents.
• An idempotnet decomposition of an idenpotent $e$, we mean $e=e_1+\cdots +e_n$ into a sum of orthogonal idenpotents. If moreover $e_1,\cdots ,e_n$ are primitive, the it is called a primitive idenpotnet decomposition.

For example, for $e\in R$ with $e^2=e$, $1=e+(1-e)$ is a idempotent decomposition.

Theorem

Let $e$ be an idempotent of $R$. If $e=e_1+\cdots +e_n$ is an idempotent decomposition, then $Re=R{e}_1\oplus \cdots \oplus R{e}_n$ as left ideals or left $R$-modules.

Conversely, if $Re$ is a direct sum of left ideals of $R$, $Re=I_1\oplus \cdots \oplus I_n$, then there is an idempotent decomposition $e=e_1+\cdots +e_n$ with $e_i\in I_i$ and $I_i=R{e}_i$.

In this way, the idempotent decomposition of $e$ are in bijection with the direct sum decomposition of the left ideals $Re$.

\begin{proof} See ^1dezlm. \end{proof}

Corollary

Let $e$ be an idem of $R$. TFAE:

• $e$ is primitive;
• ${}_R(Re)$ is indecomposable as a left $R$-module;
• $e$ is the unique idempotent in $eRe$.

\begin{proof} See ^tv7l9r. \end{proof}

Theorem

Let $e,f$ be idempotents of $R$, and let $V$ be an $R$-module.

• ${\mathrm{Hom}}_R(Re,V)\simeq eV$. In particular, ${\mathrm{Hom}}_R(Re,Rf)\simeq fRe$ as abelian groups.
• ${\mathrm{End}}_R(Re)\simeq (eRe{)}^{\mathrm{op}}$ as rings.

\begin{proof} See ^yr2ffd. \end{proof}

Theorem

Let $e,f$ be idempotent of $R$. TFAE:

• $Re\simeq Rf$ as left $R$-modules.
• $eR\simeq fR$ as right $R$-modules.
• There exist $b\in fRe$ and $a\in eRf$ with $ab=e$, $ba=f$.

\begin{proof} See ^spaio1. \end{proof}

Jacobson Radical

Definition

Let $R$ be a ring. The Jacobson radical $J(R)$ of $R$ is the intersection of all left maximal ideals of $R$. As a left $R$-module, $J(R)=\mathrm{Rad}({}_{R}R)$.

Theorem

The following holds for the radical of a ring $R$.

• $J(R)$ is the intersection of all the annihilator ideals of simple (left) $R$-modules. Consequently, $J(R)$ is a $2$-sided ideal of $R$.
• $a\in J(R)$ iff $1-xa$ has a left inverse for all $x\in R$, i.e., there exists $b\in R$ such that $b(1-xa)=1$.
• $J(R)$ is the largest one among the ideals of $I$ satisfying the following condition: if $a\in I$, then $1-a$ is a unit of $R$.
• $J(R)$ coincides with the intersection of all maximal right ideals of $R$.

\begin{proof} See ^p95eh4 \end{proof}

Definition

• An element $a\in R$ is called nilpotent if $a^n=0$ for some $n\in {\mathbb{Z}}_{+}$. If so, $(1-a)(1+a+\cdots +a^{n-1})=1$ and $1-a$ is a unit.
• An left ideal $I$ of $R$ is called a nil left ideal if every element of $I$ is nilpotent.
• $I$ is called nilpotent if $I^n=0$ for some $n$. Clearly, nilpotent ideals are nil.

Theorem

If $I$ is a nil left ideal, then $I\subseteq J(R)$.

\begin{proof} See ^112eab. \end{proof}

Theorem

If $A$ is Artinian, then $J(A)$ is nilpotent.

\begin{proof} See ^a43f1d. \end{proof}

Corollary

Let $A$ be an Artinian ring, then $J(A)$ is the maximal nilpotent left ideal of $A$.

\begin{proof} See ^8uxtcw. \end{proof}

Corollary

Let $A$ be an Artinian ring. Then $J(A)$ is the unique left ideal $U$ of $A$ with the property that $U$ is nilpotnent and $A/U$ is semisimple.

\begin{proof} See ^ds8ywb. \end{proof}

Theorem

Let $I$ be a $2$-sided ideal of $R$ with $I\subseteq J(R)$. Then $J(R/I)=J(R)/I$.

\begin{proof} See ^dh47bb. \end{proof}

Theorem

Let $A$ be an Artinian. Then $A$ is semisimple iff $J(A)=0$.

\begin{proof} See ^t0akw2. \end{proof}