结构矩阵环上的优良分次
Very good gradings on structural matrix rings
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中文总结 AI 辅助
研究结构矩阵环Mₙ(ρ,R)上的优良G分次，证明其与预序ρ上的分次在平凡、对称等性质上等价，给出epsilon-强性判据，还建立了其与G自由偏作用的一一对应关系及分类结果。
中文摘要 AI 辅助
设R为非零结合单位环，G为群，ρ为{1,…,n}上的预序。ρ上的G分次会诱导出结构矩阵环Mₙ(ρ,R)上的优良G分次。我们证明，对于平凡、对称、epsilon-强和强这四种性质，ρ上的分次具有该性质当且仅当诱导的环分次也具有该性质。epsilon-交叉积和交叉积性质可从ρ传递到环，但一般情况下反之不成立。我们还给出了epsilon-强性的具体判据，并证明Mₙ(ρ,R)上的强优良G分次满足|G|≤n。当ρ为等价关系且中性分支为对角时，优良分次与G在{1,…,n}上的自由偏作用一一对应，且轨道关系为ρ；这些分次是epsilon-交叉积，在域上该对应给出了分次代数同构下的分类。
英文摘要
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.
