Von Neumann regularity of smash products associated with G-set gradings

Abstract

For a G-graded ring R, the problem of Gr-regularity (i.e. von Neumann regularity condition on homogeneous elements) has been treated earlier (see Năstăsescu and Van Oystaeyen, 1982 [11]). Once the concept of the smash product was introduced, this problem has been resumed by several authors. This paper is concerned with von Neumann regularity of two smash products: the smash product R # G of the G-graded ring R by the group G and the smash product R # A of R by a finite left G-set A. The connections between the regularity of R # A and the (Gr-)regularity of R are also investigated. One consequence of our results is that the smash product R # A is a von Neumann regular ring if and only if the category ( G , A , R ) - gr is regular, in Stenström's sense.