In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements x, y we have[1] where |x| denotes the grade of the element and is 0 or 1 (in Z2) according to whether the grade is even or odd, respectively.
Equivalently, it is a superalgebra where the supercommutator always vanishes.
Algebraic structures which supercommute in the above sense are sometimes referred to as skew-commutative associative algebras to emphasize the anti-commutation, or, to emphasize the grading, graded-commutative or, if the supercommutativity is understood, simply commutative.
In particular, the square of any odd element x vanishes whenever 2 is invertible: Thus a commutative superalgebra (with 2 invertible and nonzero degree one component) always contains nilpotent elements.
A Z-graded anticommutative algebra with the property that x2 = 0 for every element x of odd grade (irrespective of whether 2 is invertible) is called an alternating algebra.