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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 of 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.