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: for homogeneous elements x, y in M of degree |x&hairsp;|, |y&hairsp;|. By construction, a graded-symmetric algebra is graded-commutative; i.e., $$xy = (-1)^{|x||y|} yx$$ and is universal for this.
 * $$xy - (-1)^{|x||y|}yx$$
 * $$x^2$$ when |x&hairsp;| is odd

In spite of the name, the notion is a common generalization of a symmetric algebra and an exterior algebra: indeed, if V is a (non-graded) R-module, then the graded-symmetric algebra of V with trivial grading is the usual symmetric algebra of V. Similarly, the graded-symmetric algebra of the graded module with V in degree one and zero elsewhere is the exterior algebra of V.