Superalgebra

A superalgebra over K is a K-module A with a direct sum decomposition

together with a bilinear multiplication A × A → A such that

where the subscripts are read modulo 2.

A superring, or Z₂-graded ring, is a superalgebra over the ring of integers Z.

The elements of A_i are said to be homogeneous. The parity of a homogeneous element x, denoted by |x|, is 0 or 1 according to whether it is in A₀ or A₁. Elements of parity 0 are said to be even and those of parity 1 to be odd. If x and y are both homogeneous then so is the product xy and

An associative superalgebra is one whose multiplication is associative and a unital superalgebra is one with a multiplicative identity element. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital.

A commutative superalgebra is one which satisfies a graded version of commutativity. Specifically, A is commutative if

for all homogeneous elements x and y of A.