Graded-symmetric algebra
In algebra, given a commutative ring R, the graded-symmetric algebra of a graded R-module M is the quotient of the tensor algebra of M by the ideal I generated by elements of the form: x y − ( − 1 ) | x | | y | y x {\displaystyle xy-(-1)^{|x||y|}yx} x 2 {\displaystyle x^{2}} when |x| is odd for homogeneous elements x, y in M of degree |x|, |y|. By construction, a graded-symmetric algebra is graded-commutative; i.e., x y = ( − 1 ) | x | | y | y x {\displaystyle xy=(-1)^{|x||y|}yx} and is universal for this.