Annihilator (ring theory)

Annihilators are a concept that occurs in ring theory, a branch of mathematics. Let R be a ring, and let M be a left R-module. Choose a subset S of M. The annihilator Ann_RS of S is the set of all elements r in R such that for each s in S, rs=0.

The annihilator of a single element x is usually written Ann_Rx instead of Ann_R{x}. If the ring R can be understood from the context, the subscript R is usually omitted.

Annihilators are always one-sided ideals of their ring: If a and b both annihilate S, then for each s in S, (a+b)s=as+bs=0, and for any c in R, (ca)s=c(as)=c0=0. The annihilator of M is even a two-sided ideal: (ac)s=a(cs)=0, since cs is another element of M.

M is always a faithful R/Ann_RM-module.