ON WEAKLY RIGID RINGS

A. R. NASR-ISFAHANI, A. MOUSSAVI
2009 Glasgow Mathematical Journal  
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
more » ... 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, R[x], R[x; α, δ]andR[x, x −1 ; α] 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. 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 that δ According 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 second author and E. Hashemi in [12] defined a ring R with a monomorphism α and an α-derivation δ, to be called (α, δ)-compatible if for each a, b ∈ R, ab = 0 implies aδ(b) = 0, and ab = 0 if and only if aα(b) = 0. We say a ring R with a monomorphism α and α-derivation δ, to be called (α, δ)weakly rigid if for each a, b ∈ R, aRb = 0 implies aδ(b) = 0, and aRb = 0 if and only if aα(Rb) = 0. 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. https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089509005084 Downloaded from https://www.cambridge.org/core. IP address: 207.241.231.82, on 28 Jul 2018 at 08:33:10, subject to the Cambridge Core terms of use, available at THEOREM 3.15. Let R be an α-weakly rigid ring. If R is a quasi-Baer ring, then R[x, x −1 ; α] is a quasi-Baer ring. Proof. Since R is α-weakly rigid quasi-Baer, A is quasi-Baer. Since α is an automorphism of A and R[x, x −1 ; α] A[x, x −1 ; α], so the result follows by Theorem 3.14. LEMMA 3.16. Let R be an α-weakly rigid ring and α an automorphism of R. Let L = { R (RU) | U ⊆ R}, M = { S (SU) | U ⊆ S = R[x, x −1 ; α]} and : L → M, given by (I) = I[x, x −1 ; α] and : M → L, given by (J) = J ∩ R; then o = id L . Proof. The proof is similar to that of Lemma 3.5. THEOREM 3.17. Let R be an α-weakly rigid ring and α an automorphism of R. If R[x, x −1 ; α] is quasi-Baer, then R is quasi-Baer.
doi:10.1017/s0017089509005084 fatcat:2mtrspqghre4vfldq7au2fv6au