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

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 Nǎstǎsescu 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.