Jacobson Radical Annihilates Simple Modules

Statement

Let $R$ be a commutative ring and $M$ a simple $R$-module.

Theorem.
The Jacobson radical satisfies

so in particular

Reason (standard): $\mathrm{Ann}_R(M)$ is a maximal ideal , and by J(R) = ⋂ max ideals we have $J(R)$ contained in every maximal ideal, hence in $\mathrm{Ann}_R(M)$. (See also annihilator of a module .)

This fact is one of the key inputs behind Nakayama's lemma .

Examples

1. Local ring: the maximal ideal kills the residue field.
   If $R$ is local with maximal ideal $\mathfrak m$, then $J(R)=\mathfrak m$.
   The simple $R$-module is $R/\mathfrak m$ (the residue field ), and indeed $\mathfrak m\cdot (R/\mathfrak m)=0$.

   Example: $R=k[x]/(x^2)$, $\mathfrak m=(\bar x)$. Then $\bar x$ acts by $0$ on $k\cong R/(\bar x)$.

2. Integers.
   $J(\mathbb Z)=(0)$, so $J(\mathbb Z)M=0$ for every simple $\mathbb Z$-module.
   The simple $\mathbb Z$-modules are $\mathbb Z/p\mathbb Z$, and the zero ideal acts as $0$ as expected.

3. A finite quotient of $\mathbb Z$.
   Let $R=\mathbb Z/12\mathbb Z$. Then $J(R)=(6)$ (intersection of the maximal ideals $(2)$ and $(3)$).
   The simple $R$-modules are $R/(2)\cong \mathbb Z/2\mathbb Z$ and $R/(3)\cong \mathbb Z/3\mathbb Z$.
   In both cases, the class of $6$ acts as $0$, so $J(R)$ annihilates each simple module.