modules over decomposable rings

Let $R_{1},R_{2}$ be two, nontrivial, unital rings and $R=R_{1}\oplus R_{2}$ . If $M_{1}$ is a $R_{1}$ -module and $M_{2}$ is a $R_{2}$ -module, then obviously $M_{1}\oplus M_{2}$ is a $R$ -module via $(r,s)\cdot(m_{1},m_{2})=(r\cdot m_{1},s\cdot m_{2})$ . We will show that every $R$ -module can be obtain in this way.

Proposition. If $M$ is a $R$ -module, then there exist submodules $M_{1},M_{2}\subseteq M$ such that $M=M_{1}\oplus M_{2}$ and for any $r\in R_{1}$ , $s\in R_{2}$ , $m_{1}\in M_{1}$ and $m_{2}\in M_{2}$ we have

(r,s)\cdot m_{1}=(r,0)\cdot m_{1}\ \ \ \ (r,s)\cdot m_{2}=(0,s)\cdot m_{2},

i.e. ring action on $M_{1}$ (respectively $M_{2}$ ) does not depend on $R_{2}$ (respectively $R_{1}$ ).

Proof. Let $e=(1,0)\in R$ and $f=(0,1)\in R$ . Of course both $e, f$ are idempotents and $(1,1)=e+f$ . Moreover $ef=fe=0$ and $e, f$ are central, i.e. $e,f\in\{c\in R\ \big{|}\ \forall_{x\in R}\ cx=xc\}$ . We will use $e, f$ to construct submodules $M_{1},M_{2}$ . More precisely, let $M_{1}=eM$ and $M_{2}=fM$ . Because $e, f$ are central, then it is clear that both $M_{1}$ and $M_{2}$ are submodules. We will show that $M_{1}+M_{2}=M$ . Indeed, let $m\in M$ . Then we have

m=(1,1)\cdot m=(e+f)\cdot m=e\cdot m+f\cdot m.

Thus $M_{1}+M_{2}=M$ . Furthermore, assume that $m\in M_{1}\cap M_{2}$ . Then there exist $m_{1},m_{2}\in M$ such that

e\cdot m_{1}=m=f\cdot m_{2}

and therefore

e\cdot m_{1}-f\cdot m_{2}=0.

Now, after multiplying both sides by $e$ we obtain that

0=(ee)\cdot m_{1}-(ef)\cdot m_{2}=e\cdot m_{1}-0\cdot m_{2}=e\cdot m_{1}=m,

thus $M_{1}\cap M_{2}=0$ . This shows that $M=M_{1}\oplus M_{2}$ . To finish the proof, we need to show that the ring action on $M_{1}$ does not depend on $R_{2}$ (the other case is analogous). But this is clear, since for any $(r,s)\in R$ and $m\in M$ we have

(r,s)\cdot(e\cdot m)=\big{(}(r,s)(1,0)\big{)}\cdot m=(r,0)\cdot m=\big{(}(r,0)% (1,0)\big{)}\cdot m=(r,0)\cdot(e\cdot m).

This completes the proof. $\square$

Title	modules over decomposable rings
Canonical name	ModulesOverDecomposableRings
Date of creation	2013-03-22 18:50:05
Last modified on	2013-03-22 18:50:05
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	4
Author	joking (16130)
Entry type	Theorem
Classification	msc 20-00
Classification	msc 16-00
Classification	msc 13-00