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
![{\displaystyle xy=(-1)^{|x||y|}yx,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88344ec6b363e90760ae7c5e6aacb625fb8d1c5f)
where |x | and |y | denote the degrees of x and y.
A commutative (non-graded) ring, with trivial grading, is a basic example. For example, an exterior algebra is generally not a commutative ring but is a graded-commutative ring.
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.
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