Annihilator (ring theory)

In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of S. Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator. The above definition applies also in the case noncommutative rings, where the left annihilator of a left module is a left ideal, and the right-annihilator, of a right module is a right ideal. Let R be a ring, and let M be a left R-module. Choose a non-empty subset S of M. The annihilator of S, denoted AnnR(S), is the set of all elements r in R such that, for all s in S, rs = 0. In set notation, for all It is the set of all elements of R that "annihilate" S (the elements for which S is a torsion set). Subsets of right modules may be used as well, after the modification of "sr = 0" in the definition. The annihilator of a single element x is usually written AnnR(x) instead of AnnR({x}). If the ring R can be understood from the context, the subscript R can be omitted. Since R is a module over itself, S may be taken to be a subset of R itself, and since R is both a right and a left R module, the notation must be modified slightly to indicate the left or right side. Usually and or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary. If M is an R-module and AnnR(M) = 0, then M is called a faithful module. If S is a subset of a left R module M, then Ann(S) is a left ideal of R. If S is a submodule of M, then AnnR(S) is even a two-sided ideal: (ac)s = a(cs) = 0, since cs is another element of S. If S is a subset of M and N is the submodule of M generated by S, then in general AnnR(N) is a subset of AnnR(S), but they are not necessarily equal. If R is commutative, then the equality holds. M may be also viewed as an R/AnnR(M)-module using the action . Incidentally, it is not always possible to make an R module into an R/I module this way, but if the ideal I is a subset of the annihilator of M, then this action is well-defined.