Von Neumann regularity of smash products associated with \(G\)-set gradings.
Let \(G\) be a multiplicative group. Let \(R\) be a \(G\)-graded ring and \(A\) a finite left \(G\)-set. The smash product \(R\#A\) is the free left \(R\)-module with basis \(\{p_x\mid x\in A\}\), and multiplication defined by \((r_\sigma p_x)(s_\tau p_y)=(r_\sigma s_\tau)p_y\) if \(\tau y=x\) and zero otherwise, for any \(r_\sigma\in R_\sigma\), \(s_\tau\in R_\tau\) and \(x,y\in A\). The smash product \(R\#A\) is a \(G\)-graded ring with its \(\sigma\)-homogeneous component \((R\#A)_\sigma=\sum_{x\in A}R_\sigma p_x\) for each \(\sigma\in G\). The smash product is a useful tool in the theory of graded rings. A ring \(R\) is said to be von Neuman regular if for any \(r\in R\) there exists \(s\in R\) such that \(r=rsr\). Von Neumann regular rings play an important role in ring theory. This paper is devoted to obtain conditions under which a smash product \(R\#A\) of a \(G\)-graded ring \(R\) and a finite left \(G\)-set \(A\) is von Neumann regular.
scientific article; zbMATH DE number 5949281
scientific article