Editing Annihilator (ring theory) (section)

{{Short description|Ideal that maps to zero a subset of a module}}
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In [[mathematics]], the '''annihilator''' of a [[subset]] {{mvar|S}} of a [[module (mathematics)|module]] over a [[ring (mathematics)|ring]] is the [[ideal (ring theory)|ideal]] formed by the elements of the ring that give always zero when multiplied by each element of {{mvar|S}}. 

Over an [[integral domain]], a module that has a nonzero annihilator is a [[torsion module]], and a [[finitely generated module|finitely generated]] torsion module has a nonzero annihilator.

The above definition applies also in the case of [[Noncommutative ring|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.