On weakly rigid rings

Abstract

Let R be a ring with a monomorphism α and an α-derivation δ. We introduce (α,δ)-weakly rigid rings which are a generalisation of α-rigid rings and investigate their properties. Every prime ring R is (α,δ)-weakly rigid for any automorphism α and α-derivation δ. It is proved that for any n, a ring R is (α,δ)-weakly rigid if and only if the n-by-n upper triangular matrix ring Tn(R) is (ᾱ,δ̄ )-weakly rigid if and only if Mn(R) is (ᾱ,δ̄)-weakly rigid. Moreover, various classes of (α,δ )-weakly rigid rings is constructed, and several known results are extended. We show that for an (α,δ )-weakly rigid ring R, and the extensions R[x], R[[x]], R[x;α,δ ], R[x, x-1; ], R[[x;α ]], R[[x, x -1;α ]], the ring R is quasi-Baer if and only if the extension over R is quasi-Baer. It is also proved that for an (α,δ)-weakly rigid ring R, if any one of the rings R, R[x], R[x;α,δ ] and R[x, x-1;α ] is left principally quasi-Baer, then so are the other three. Examples to illustrate and delimit the theory are provided. © 2009 Glasgow Mathematical Journal Trust.