Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

给定环A上的逆半群S的动作 ${\varphi }$ （ ${\varphi }(s)$ 的定义域 由 $D_{s^*}$ 表示 ），我们证明如果理想 $D_e $ ，具有e幂等性，是单位的，那么偏斜逆半群环 $A\rtimes S$ 可以实现为一个充足的群群的卷积代数，其系数在一组（单位）环中。相反，我们证明了在一组环中具有系数的充足群群的卷积代数与这种偏斜逆半群环同构。我们恢复了文献中关于一个领域的 Steinberg 代数的已知结果作为特殊情况。