(Jordan) derivation on amalgamated duplication of a ring along an ideal

Let A be a ring and I be an ideal of A. The amalgamated duplication of A along I is the subring of A × A defined by $A\bowtie I := {(a, a + i) |a ∈ A, i ∈ I}$. In this paper, we characterize $A\bowtie I$ over which any (resp. minimal) prime ideal is invariant under any derivation provided that A is semiprime. When A is noncommutative prime, then $A\bowtie I$ is noncommutative semiprime (but not prime except if I = (0)). In this case, we prove that any map of $A\bowtie I$ which is both Jordan and Jordan triple derivation is a derivation.

Rings with S-acc on d-annihilators

A commutative ring [Formula: see text] is said to satisfy acc on d-annihilators if for every sequence [Formula: see text] of elements of [Formula: see text] the sequence [Formula: see text] is stationary. In this paper we extend the notion of rings with acc on d-annihilators by introducing the concept of rings with [Formula: see text]-acc on d-annihilators, where [Formula: see text] is a multiplicative set. Let [Formula: see text] be a commutative ring and [Formula: see text] a multiplicative subset of [Formula: see text] We say that [Formula: see text] satisfies [Formula: see text]-acc on d-annihilators if for every sequence [Formula: see text] of elements of [Formula: see text] the sequence [Formula: see text] is [Formula: see text]-stationary, that is, there exist a positive integer [Formula: see text] and an [Formula: see text] such that for each [Formula: see text] [Formula: see text] We give equivalent conditions for the power series (respectively, polynomial) ring over an Armendariz ring to satisfy [Formula: see text]-acc on d-annihilators. We also study serval properties of rings satisfying [Formula: see text]-acc on d-annihilators. The concept of the amalgamated duplication of [Formula: see text] along an ideal [Formula: see text] [Formula: see text] is studied.

Radical factorization for trivial extensions and amalgamated duplication rings

Let [Formula: see text] be a commutative ring extension such that [Formula: see text] is a trivial extension of [Formula: see text] (denoted by [Formula: see text]) or an amalgamated duplication of [Formula: see text] along some ideal of [Formula: see text] (denoted by [Formula: see text]. This paper examines the transfer of AM-ring, N-ring, SSP-ring and SP-ring between [Formula: see text] and [Formula: see text]. We study the transfer of those properties to trivial ring extension. Call a special SSP-ring an SSP-ring of the following type: it is the trivial extension of [Formula: see text] by a C-module [Formula: see text], where [Formula: see text] is an SSP-ring, [Formula: see text] a von Neumann regular ring and [Formula: see text] a multiplication C-module. We show that every SSP-ring with finitely many minimal primes which is a trivial extension is in fact special. Furthermore, we study the transfer of the above properties to amalgamated duplication along an ideal with some extra hypothesis. Our results allows us to construct nontrivial and original examples of rings satisfying the above properties.

C∗-algebras defined by amalgamated duplication of C∗-algebras

Let [Formula: see text] and [Formula: see text] be two [Formula: see text]-algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible action of [Formula: see text] on [Formula: see text]. We define [Formula: see text] as a [Formula: see text]-algebra, where it is a strongly splitting [Formula: see text]-algebra extension of [Formula: see text] by [Formula: see text]. Normal, self-adjoint, unitary, invertible and projection elements of [Formula: see text] are characterized; sufficient and necessary conditions for existing unit and bounded approximate identity of [Formula: see text] as a Banach algebra and as a [Formula: see text]-algebra are given. We characterize ∗-automorphisms on [Formula: see text] and give some results related to ∗-homomorphisms, ∗-representations and completely bounded maps on this [Formula: see text]-algebra. Also, we have constructed a new Hilbert [Formula: see text]-module [Formula: see text] over [Formula: see text], where [Formula: see text] is a Hilbert [Formula: see text]-module over [Formula: see text] and [Formula: see text] is a Hilbert [Formula: see text]-module over [Formula: see text].

Biprojectivity and biflatness of amalgamated duplication of Banach algebras

Let [Formula: see text] and [Formula: see text] be two Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible action of [Formula: see text] on [Formula: see text]. Let [Formula: see text] be a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. We show that (super) amenability of [Formula: see text] implies (super) module amenability of [Formula: see text] and (super) amenability [Formula: see text]. We investigate biprojectivity and biflatness of [Formula: see text] in the some especial cases. We also give some results related to module biprojectivity and module biflatness of [Formula: see text], when [Formula: see text] is biprojective or biflat.

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

Gorenstein dimensions over some rings of the form R ⊕ C

Given a semidualizing module [Formula: see text] over a commutative Noetherian ring, Holm and Jørgensen [Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205(2) (2006) 423–445] investigate some connections between [Formula: see text]-Gorenstein dimensions of an [Formula: see text]-complex and Gorenstein dimensions of the same complex viewed as a complex over the “trivial extension” [Formula: see text]. We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D’Anna and Fontana’s amalgamated duplication [An amalgamated duplication of a ring along an ideal: The basic properties, J. Algebra Appl. 6(3) (2007) 443–459] and Enescu’s pseudocanonical cover [A finiteness condition on local cohomology in positive characteristic, J. Pure Appl. Algebra 216(1) (2012) 115–118].