I am self studying ring theory and I came upon the concept of center.
The center of a ring $R$ is a subring $Z(R)$ such that $xy=yx$ for $x \in Z(R), y \in R$.
I am wondering if there is also a concept for the case where $xy = -yx$, i.e. an anticommutative counterpart of a center. I will appreciate references for which this case is discussed as I think this will be valuable to what I am working on for research (mathematical physics).
Let $R$ be a $\Bbb Z$-graded ring. For every $n\in\Bbb Z$ let $R_n$ denote the subgroup of $R$ consisting of $0$ and all elements of degree $n$. For every $n\in\Bbb Z$ let $$Z_n=\{x\in R_n:\forall m\in\Bbb Z\forall y\in R_m(xy=(-1)^{nm}yx)\}$$ and $Z=\sum_nZ_n$. Then $Z$ is a graded subring of $R$.