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 $b\in R$:

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

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
Canonical name	Supercommutative
Date of creation	2013-03-22 13:25:18
Last modified on	2013-03-22 13:25:18
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	7
Author	rmilson (146)
Entry type	Definition
Classification	msc 16W50
Synonym	graded-commutative
Synonym	supercommutative algebra
Synonym	even element
Synonym	odd element
Related topic	SuperAlgebra