On Weakly Rigid Rings

Abstract

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 T n (R) is (ᾱ,δ)-weakly rigid if and only if M n (R) is (ᾱ,δ)-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, 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,is left principally quasi-Baer, then so are the other three. Examples to illustrate and delimit the theory are provided.AMS Subject Classification. 16S36. 16W60.1. Introduction. Throughout this paper R denotes an associative ring with unity; α is a monomorphism of R which is not assumed to be surjective; and δ an α-derivation of R, that is δ is an additive map such thatAccording to Krempa [18], a monomorphism α of a ring R is called to be rigid if aα(a) = 0 implies a = 0 for a ∈ R. A ring R is said to be α-rigid if there exists a rigid monomorphism α of R.The By [12], a ring R is α-rigid if and only if it is (α, δ)-compatible and reduced. Notice that the class of α-rigid rings and (α, δ)-compatible rings is a narrow class of rings, and it is easy to see that every (α, δ)-compatible ring is (α, δ)-weakly rigid; but there are various classes of (α, δ)-weakly rigid rings which are not (α, δ)-compatible, as we will see in Section 2.It is clear that every prime ring R is (α, δ)-weakly rigid for any automorphism α and α-derivation δ. In this paper we prove that for any positive integer n, a ring R is (α, δ)-weakly rigid if and only if the n-by-n upper triangular matrix ring T n (R) is (ᾱ,δ)-weakly rigid if and only if the matrix ring M n (R) is (ᾱ,δ)-weakly rigid.

On NI Skew Polynomial Rings

“…According to Krempa [5], an endomorphism of a ring R is called rigid if a a = 0 implies a = 0 for any a ∈ R. Such an endomorphism is then automatically a monomorphism. A ring R is said to be -rigid if there exists a rigid monomorphism of R (for more details about -rigid rings and its generalizations see [4], [5] and [9]). We say that a subset I ⊆ R is -rigid if for any a ∈ R, a a ∈ I implies that a ∈ I.…”

On Annihilations of Ideals in Skew Monoid Rings

“…According to [41], a ring R with a monomorphism α is called α-weakly rigid if for each a, b ∈ R, aRb = 0 if and only if aα(Rb) = 0. For any positive integer n, a ring R is α-weakly rigid if and only if, the n-by-n upper triangular matrix ring T n (R) is α-weakly rigid if and only if, the matrix ring M n (R) is α-weakly rigid, where α((a ij )) = (α(a ij )) for each (a ij ) ∈ M n (R).…”

The ascending chain condition for principal left or right ideals of skew generalized power series rings

“…Now we examine the ACCPL condition for the skew Laurent polynomial ring R[x, x −1 ; α]. First, we recall the following propositions which are proved in [16,24].…”