# Thread: Annihilators and torsion R-Modules

1. ## Annihilators and torsion R-Modules

Let R be a PID, and let B be a torsion R-module, and let p be a prime in R.

Prove that if pb=0 for b in B, then Ann(B) is in (p), the ideal generated by p.

2. Originally Posted by robeuler
Let R be a PID, and let B be a torsion R-module, and let p be a prime in R.

Prove that if pb=0 for b in B, then Ann(B) is in (p), the ideal generated by p.
The Ann(B) is defined as

$\text{Ann(B)} = \{ r \in R | rb = 0, \forall b \in B\}$,

(T(B)=B since B is a torsion R-module, where T(B) is a torsion submodule of B).

We see that Ann(B) is an ideal of R.

Since R is a PID, every non-zero prime ideal is a maximal ideal. Every ideal in R (except R itself) is contained in a maximal ideal. Thus, Ann(B) is contained in a maximal ideal.

Since pb=0 for b in B, we have $p \in Ann(B)$ by the definition of an annhilator. We also have $p \in (p)$, where p is a prime. Since (p) is a maximal ideal, any ideal I(except R itself) containing an element of (p) should be contained in (p). Thus, Ann(B) is in (p).