On Quotient Rings of Trivial Extensions

On derivations and biderivations of trivial extensions and triangular matrix rings

‎Triangular matrix rings are examples of trivial extensions‎. ‎In this article we determine the structure of derivations and biderivations of the trivial extensions‎, ‎and thereby we describe the derivations and biderivations of the upper triangular matrix rings‎. ‎Some related results are also obtained‎.

Gaussian trivial ring extensions and fqp-rings

Let A be a commutative ring and E a non-zero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are cha...

Trivial Extensions of Local Rings and a Conjecture of Costa

This paper partly settles a conjecture of Costa on (n, d)-rings, i.e., rings in which n-presented modules have projective dimension at most d. For this purpose, a theorem studies the transfer of the (n, d)-property to trivial extensions of local rings by their residue fields. It concludes with a brief discussion -backed by original examplesof the scopes and limits of our results. ∗This project ...

Extensions of Nilpotent P.P. Rings

Extensions of strongly alpha-reversible rings

We introduce the notion ofstrongly $alpha$-reversible rings which is a strong version of$alpha$-reversible rings, and investigate its properties. We firstgive an example to show that strongly reversible rings need not bestrongly $alpha$-reversible. We next argue about the strong$alpha$-reversibility of some kinds of extensions. A number ofproperties of this version are established. It is shown ...