Prove that an ideal $ \mathfrak{m} $ of a commutative ring $ R $ is maximal iff $ R/\mathfrak{m} $ is simple.

Question

Could someone give me a hint on whether I’m on the right track or not? For sufficiency, I tried the following:

Suppose that $ \mathfrak{m} $ is a maximal ideal. With the quotient map, we get $ R/\mathfrak{m} $, where any $ x \in R/\mathfrak{m} $ is of the form $ a + \mathfrak{m} $, with $ a \in R $. Suppose that $ J $ is a non-zero ideal of $ R/\mathfrak{m} $, and let $ y \in J $. As $ J $ is an ideal, we have $ x y \in J $, so $ x y = a y + \mathfrak{m} $. Given that $ a \in R $, $ y \in J $ and $ J \subseteq R $, we get $ y \in R $, so $ a y \in R $. This implies that $ a y + \mathfrak{m} \in R/\mathfrak{m} $. Hence, $ x y \in J $ and $ x y \in R/\mathfrak{m} $, and so $ J \subseteq R/\mathfrak{m} $.

Now, pick any $ a b + \mathfrak{m} \in R/\mathfrak{m} $, where $ b \in J $, with $ J $ an ideal of $ R/\mathfrak{m} $. We know that $$ a b + \mathfrak{m} = (a + \mathfrak{m}) b \in R/\mathfrak{m}, $$ and as $ b \in J $, we have $ (a + \mathfrak{m}) b \in J $. Hence, $ R/\mathfrak{m} \subseteq J $. We have therefore shown that $ J = R/\mathfrak{m} $ if $ J $ is not the zero ideal.

We thus have only two cases, (i) when $ J $ is not the zero ideal, which we showed implies $ J = R/\mathfrak{m} $, and (ii) when $ J $ is the zero ideal. Therefore, these are the only ideals of $ R/\mathfrak{m} $, and so $ R/\mathfrak{m} $ is simple.

Is this proof on the right path? Also, does anyone have a hint for necessity?

Answer 1

This is wrong because you can't multiply an element of $R/M$ by an element of $J$. They're separate rings. The proof of this I'm familiar with supposes $M$ is maximal and constructs the ideal

$$J=\{m+rx:m\in M,r\in R\}$$

where $x$ is an element such that $x+M\neq M$ (or $x\notin M$). Then this ideal contains $M$ by letting $r=0$, and it contains $x$ by letting $m=0,r=1$. Therefore $M\subset J\implies J=R$. Then we know $1\in J$. So $1=m+rx$ for $m\in M,r\in R$.

I'll let you try to figure out the rest

Answer 2

These are all standard results:

$(1)$ Given an $R$-module $N$ and a submodule $N'$, there is a canonical bijection between the collection of submodules of $N''=N/N'$ and the collection of submodules $M$ of $N$ such that $M\supseteq N'$.

$(2)$ A module $S$ is simple if it has precisely two distinct submodules.

$(3)$ Finally, an ideal $\mathfrak m$ is maximal if and only if the only ideals lying over $\mathfrak m$ are $\mathfrak m$ and the unit ideal $R$.

Answer 3

If $\mathfrak m$ is maximal, then $R/\mathfrak m$ is a field and hence it is simple.

Conversely, if $R/\mathfrak m$ is simple, then by the Correspondence Theorem, $\mathfrak m$ is maximal.