# Graded-commutative ring

In algebra, a **graded-commutative ring** (also called a **skew-commutative ring**) is a graded ring that is commutative in the graded sense; that is, homogeneous elements *x*, *y* satisfy

where |*x*|, |*y*| denote the degrees of *x*, *y*.

A commutative (non-graded) ring, with trivial grading, is a basic example. An exterior algebra is an example of a graded-commutative ring that is not commutative in the non-graded sense.

A cup product on cohomology satisfies the skew-commutative relation; hence, a cohomology ring is graded-commutative. In fact, many examples of graded-commutative rings come from algebraic topology and homological algebra.

## ReferencesEdit

- David Eisenbud,
*Commutative Algebra. With a view toward algebraic geometry*, Graduate Texts in Mathematics, vol 150, Springer-Verlag, New York, 1995. ISBN 0-387-94268-8 - Beck, Kristen A.; Sather-Wagstaff, Sean (2013-07-01). "A somewhat gentle introduction to differential graded commutative algebra". arXiv:1307.0369 [math.AC].

## See alsoEdit

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