Search Results for “graded commutative ring”

Source: 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
=
(
−
1

)

|

x

|

|

y

|

y
x
,

{\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 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.