I am solving an exercise problem which aims to prove $R$-modules $Ra$ and $R/\mathrm{Ann}_R(a)$ are isomorphic. This exercise problem is broken down to steps.
Let $R$ be a ring and $M$ be a (left) $R$-module. Let $a\in M$. Define the annihilator of $a$ in $R$ by $\mathrm{Ann}_R(a):=\{ r\in R\mid ra=0 \}.$
Show that the submodule of $M$ generated by $a$ is $Ra:= \{ra\mid r\in R\}$.
Show that $\mathrm{Ann}_R(a)$ is a left ideal of $R$.
Show that the $R$-modules $Ra$ and $R/\mathrm{Ann}_R(a)$ are isomorphic.
Show, by example, that $\mathrm{Ann}_R(a)$ need not be a two-sided ideal in $R$.
I proved $\mathrm{Ann}_R(a)$ is a left ideal of $R$. However unable to prove the first and the third one. Can anyone give me hints or any ideas? Thanks!
Best Answer
Let $R\to Ra$ be the map sending $r\mapsto ra$. You can check that this is an $R$-module homomorphism. This map is surjective since every element of $Ra$ is of the form $ra$ for some $r\in R$ by definition. The kernel of this map is precisely $Ann_R(a)$, so by the first isomorphism theorem we get $R/Ann_R(a)\cong Ra$.
You can also define the isomorphism explicitly, but that can be deduced from the proof of the isomorphism theorem.