Let be an abelian group and be a collection of endomorphisms of . The commutant of is the set of all endomorphisms of commuting with every element of ; symbolically,
The commutant of is equal to the commutant of the subring of generated by the , so we may assume without loss of generality that is already such a subring. In that case, is just the ring of endomorphisms of as a left -module. The use of the term commutant instead can be thought of as emphasizing the role of and de-emphasizing the role of .
The assignment is a contravariant Galois connection on the lattice of subsets of , so the double commutant may be thought of as a closure operator. Today we will prove a basic but important theorem about this operator.