Annihilator (ring theory)

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Annihilators are a concept that occurs in ring theory, a branch of mathematics.

Definition

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.

Relations to other properties of rings

• The set of (left) zero divisors D_S of S can be written as

<math>D_S = \bigcup_{x \in S,\, x \neq 0}{\mathrm{Ann}_R\,x}.</math>

In particular D is the set of (left) zero divisors of R when S = R and R acts on itself as a left R-module.