¡Hi everyone!
I have a question, let me introduce the problem:
"Let be $M$ a right $R$-module, and $I$ ideal of $R$, if $I \subseteq Ann(M)$. Show $M$ has a natural structure like right $(R/I)$-module."
My two questions is:
- What means "Natural structure"
- Why i need $I \subseteq Ann(M)$?
Thanks.
It means there's some "obvious" way to make $M$ into an $R/I$-module. Let's try this and see what happens:
If we want an $R/I$-module structure on $M$, we need to define $\bar a\cdot x$ for any $\bar a\in R/I$ and $x\in M$. An obvious guess might be to define $$\bar a\cdot x:=a\cdot x\in M,$$
where $a\in R$ is some lift of $\bar a$ and $a\cdot x$ comes from the action of $R$ on $M$.
Now you should try to show, first and foremost, that this action is actually well defined. This is where you will use the assumption $I\subseteq Ann(M)$. The rest of the steps in showing it is a module structure should be pretty straightforward.