A Module $M$ is Irreducible if and only if $M$ is isomorphic to $R/I$ for a Maximal Ideal $I$.
Problem 449
Let $R$ be a commutative ring with $1$ and let $M$ be an $R$-module.
Prove that the $R$-module $M$ is irreducible if and only if $M$ is isomorphic to $R/I$, where $I$ is a maximal ideal of $R$, as an $R$-module.
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Definition (Irreducible module).
An $R$-module $M$ is called irreducible if $M$ is not the zero module and $0$ and $M$ are the only submodules of $M$.
An irreducible module
is also called a simple module.
Proof.
$(\implies)$ Suppose that $M$ is an irreducible $R$-module.
Then by definition of an irreducible module, $M$ is not the zero module.
Take any nonzero element $m\in M$, and consider the cyclic submodule $(m)=Rm$ generated by $m$.
Since $M$ is irreducible, we must have $M=Rm$.
Now we define a map $f:R\to M$ by sending $r\in R$ to $f(r)=rm$.
Then the map $f$ is an $R$-module homomorphism regarding $R$ is an $R$-module.
In fact, we have
\begin{align*}
&f(r+s)=(r+s)m=rm+sm=f(r)+f(s)\\
&f(rs)=(rs)m=r(sm)=rf(s)
\end{align*}
for any $r, s\in R$.
Since $M=Rm$, the homomorphism $f$ is surjective.
Thus, by the first isomorphism theorem, we obtain
\[R/I\cong M,\]
where $I=\ker(f)$.
It remains to show that $I$ is a maximal ideal of $R$.
Suppose that $J$ is an ideal such that $I\subset J \subset R$.
Then by the third isomorphism theorem for rings, we know that $J/I$ is an ideal of the ring $R/I\cong M$, hence $J/I$ is a submodule.
Since $M$ is irreducible, we must have either $J/I=0$ or $J/I=M$.
This implies that $J=I$ or $J=M$.
Hence $I$ is a maximal ideal.
$(\impliedby)$ Suppose now that $M\cong R/I$ for some maximal ideal $I$ of $R$.
Let $N$ be any submodule of $R/I$. (We identified $M$ and $R/I$ by the above isomorphism.)
Then $N$ is an ideal of $R/I$ since $N$ is an abelian group and closed under the action of $R$, hence that of $R/I$.
Since $R$ is a commutative ring and $I$ is a maximal ideal of $R$, we know that $R/I$ is a field.
Thus, only the ideals of $R/I$ are $0$ or $R/I$.
Hence we have $N=0$ or $N=R/I=M$.
This proves that $M$ is irreducible.
Related Question.
A similar technique in the proof above can be used to solve the following problem.
Prove that a nonzero $R$-module $M$ is irreducible if and only if $M$ is a cyclic module with any nonzero element as its generator.
See the post “A module is irreducible if and only if it is a cyclic module with any nonzero element as generator” for a proof.
Add to solve later
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[…] For a proof of this problem, see the post “A Module $M$ is Irreducible if and only if $M$ is isomorphic to $R/I$ for a Maximal Ideal $I$.“. […]