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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.
Any commutative algebra is a supercommutative algebra if given the trivial gradation (i.e. all elements are even). Grassmann algebras (also known as exterior algebras) are the most common examples of nontrivial supercommutative algebras. The supercenter of any superalgebra is the set of elements that supercommute with all elements, and is a supercommutative algebra.
for odd x and y. 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.