Knowlpedia

Annihilator of a module

The ideal of scalars that kill the entire module.

Let MM be a left RR-. The annihilator of MM is

ann⁡R(M)={r∈R:rM=0}.\operatorname{ann}_R(M)=\{r\in R: rM=0\}.

It equals the of the elementwise annihilators:

ann⁡R(M)=⋂m∈Mann⁡R(m),\operatorname{ann}_R(M)=\bigcap_{m\in M}\operatorname{ann}_R(m),

where ann⁡R(m)\operatorname{ann}_R(m) is the . For a left module, ann⁡R(M)\operatorname{ann}_R(M) is a , since it is stable under multiplication on both sides by arbitrary ring elements.

Remarks

The annihilator measures faithfulness: MM is faithful iff ann⁡R(M)=0\operatorname{ann}_R(M)=0.

Examples
  • For a commutative ring RR and ideal II, the module R/IR/I satisfies ann⁡R(R/I)=I\operatorname{ann}_R(R/I)=I.
  • If M=0M=0, then ann⁡R(M)=R\operatorname{ann}_R(M)=R.
  • If M=RM=R as a left module over itself, then ann⁡R(M)=0\operatorname{ann}_R(M)=0.