Annihilator (ring theory)
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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 AnnRS 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 AnnRx instead of AnnR{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/AnnRM-module.