Properties
If S is a subset of a left R module M, then Ann(S) left ideal of R. The proof is straightforward: If a and b both annihilate S, then for each s in S, (a + b)s = as + bs = 0, and for any r in R, (ra)s = r(as) = r0 = 0. (A similar proof follows for subsets of right modules to show that the annihilator is a right ideal.)
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 it is easy to check that equality holds.
M may be also viewed as a 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. Considered as an R/AnnR(M)-module, M is automatically a faithful module.
Read more about this topic: Annihilator (ring Theory)
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