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"; here the ideal "I" is generated by elements of the form:
for homogeneous elements "x", "y" in "M" of degree |"x"|, |"y"|. By construction, a graded-symmetric algebra is graded-commutative; i.e., formula_3 and is universal for this.
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".