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
x
y
=
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−
1
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x
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y
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y
x
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{\displaystyle xy=(-1)^{|x||y|}yx,}
where |x | and |y | denote the degrees of x and y .
A commutative (non-graded) ring , with trivial grading, is a basic example. For a nontrivial 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 .
