论文标题
在群体级别的von Neumann常规戒指和Brandt Grobles级的Leavitt Path代数
On groupoid graded von Neumann regular rings and a Brandt groupoid graded Leavitt path algebras
论文作者
论文摘要
令$ s $为部分群体，即具有部分二进制操作的集合。如果$ s $ r $ $ x $ x \ in xrx $ in xrx $的每个同质元素$ x \ in R中的$ s $ r $。如果戒指是$ s $的von neumann常规，并且如果$ s $是取消的，那么$ s $，则$ s $在s中的每个$ s \ in s $ s $中存在$ s^{ - 1} \ in s $ and idempotent elements $ e，$ e，$ $ e，$ $ f \ in s $ in $ es $ eS $ eS = sf = sf = s $ = s $ = s $ = s $ = $ fs^{ - 1} = s^{ - 1} e = s^{ - 1}，$ $ ss^{ - 1} = e $ and $ s^{ - 1} s = f，$，这是Brandt Groupoid享有的属性。我们观察到一个任意非零环的leavitt路径代数，是一个由Unitor环上的指向图形的图形，是一个由Brandt groupoid在添加剂$ \ mathbb {z}上分级的环，$，我们证明它是von Neumann的定期，并且仅在von noumann Quartial neumann Quonical neumann Quance for Cannical上获得CANNICE，因此它是常规的。 $ \ mathbb {z} $ - Leavitt Path代数的分级。
Let $S$ be a partial groupoid, that is, a set with a partial binary operation. An $S$-graded ring $R$ is said to be graded von Neumann regular if $x\in xRx$ for every homogeneous element $x\in R.$ Under the assumption that $S$ is cancellative, we characterize $S$-graded rings which are graded von Neumann regular. If a ring is $S$-graded von Neumann regular, and if $S$ is cancellative, then $S$ is such that for every $s\in S,$ there exist $s^{-1}\in S$ and idempotent elements $e,$ $f\in S$ for which $es=sf=s,$ $fs^{-1}=s^{-1}e=s^{-1},$ $ss^{-1}=e$ and $s^{-1}s=f,$ which is a property enjoyed by Brandt groupoids. We observe a Leavitt path algebra of an arbitrary non-null directed graph over a unital ring as a ring graded by a Brandt groupoid over the additive group of integers $\mathbb{Z},$ and we prove that it is graded von Neumann regular if and only if its coefficient ring is von Neumann regular, thus generalizing the recently obtained result for the canonical $\mathbb{Z}$-grading of Leavitt path algebras.
