# supercommutative

Let $R$ be a $\mathbb{Z}_{2}$-graded ring (or more generally, an associative algebra). We say that $R$ is supercommutative if for any homogeneous elements $a$ and $b\in R$:

 $ab=(-1)^{\deg a\deg b}ba.$

In other words, even homogeneous elements are in the center of the ring, and odd homogeneous elements anti-commute.

Common examples of supercommutative rings are the exterior algebra of a module over a commutative ring (in particular, a vector space) and the cohomology ring of a topological space (both with the standard grading by degree reduced mod 2).

Title supercommutative Supercommutative 2013-03-22 13:25:18 2013-03-22 13:25:18 rmilson (146) rmilson (146) 7 rmilson (146) Definition msc 16W50 graded-commutative supercommutative algebra even element odd element SuperAlgebra