Very good gradings on structural matrix rings
Let $R$ be a nonzero associative unital ring, let $G$ be a group, and let $\rho$ be a preorder on $\{1,\ldots,n\}$. A $G$-grading on $\rho$ induces a very good $G$-grading on the structural matrix ring $M_n(\rho,R)$. We show that, for each of the properties trivial, symmetric, epsilon-strong and strong, the grading on $\rho$ has the property if and only if the induced ring grading does. The epsilon-crossed product and crossed product properties pass from $\rho$ 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 $M_n(\rho,R)$ that is strong satisfies $|G|\leq n$. When $\rho$ 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 $\rho$. These gradings are epsilon-crossed products, and over a field the correspondence gives a classification up to graded algebra isomorphism.