Very good gradings on structural matrix rings

Abstract

Let $R$ be a nonzero associative unital ring, let $G$ be a group, and let $ρ$ be a preorder on $\{1,\ldots,n\}$. A $G$-grading on $ρ$ induces a very good $G$-grading on the structural matrix ring $M_n(ρ,R)$. We show that, for each of the properties trivial, symmetric, epsilon-strong and strong, the grading on $ρ$ has the property if and only if the induced ring grading does. The epsilon-crossed product and crossed product properties pass from $ρ$ to the ring, but the converses fail in general. We also give a concrete criterion for epsilon-strongness and show that a very good $G$-grading on $M_n(ρ,R)$ that is strong satisfies $|G|\leq n$. When $ρ$ is an equivalence relation and the neutral component is diagonal, very good gradings correspond bijectively to free partial actions of $G$ on $\{1,\ldots,n\}$ with orbit relation $ρ$. These gradings are epsilon-crossed products, and over a field the correspondence gives a classification up to graded algebra isomorphism.