Let A denote an R-algebra, so that A is a vector space over R and A×A->A (x, y)↦x·y. Now define Z congruent {x element A:x·y = 0 for some y element A!=0}, where 0 element Z. An Associative R-algebra is commutative if x·y = y·x for all x, y element A. Similarly, a ring is commutative if the multiplication operation is commutative, and a Lie algebra is commutative if the commutator [A, B] is 0 for every A and B in the Lie algebra.

Abelian group | abstract algebra | commutative | commutative ring