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 :<math>xy = (-1)^{|x||y|} yx,</math> 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<!-- majority? --> examples of graded-commutative rings come from algebraic topology and homological algebra.
== References == * 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}} *{{Cite arXiv |last1=Beck |first1=Kristen A. |last2=Sather-Wagstaff |first2=Keri Ann |author2-link=Keri Sather-Wagstaff|date=2013-07-01 |title=A somewhat gentle introduction to differential graded commutative algebra |eprint=1307.0369 |class=math.AC}}
== See also == *DG algebra *graded-symmetric algebra *alternating algebra *supercommutative algebra
Category:Abstract algebra
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