PS-modules over Rings of Generalized Power Series

Let R be a commutative ring and (S, ≤) a strictly totally ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. In this paper we show that if RM is a PS-module, then the module [[MS≤]] of generalized power series over M is a PS [[RS,≤]]-module.

Nilpotent Elements and Nil-Reflexive Property of Generalized Power Series Rings

Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.

Principal Quasi-Baerness of Rings of Generalized Power Series

Let R be a ring such that all left semicentral idempotents are central and （S, ≤） a strictly totally ordered monoid satisfying that 0 ≤s for all s ∈S. It is shown that [[R^S≤]], the ring of generalized power series with coefficients in R and exponents in S, is right p.q.Baer if and only if R is right p.q.Baer and any S-indexed subset of I（R） has a generalized join in I（R）, where I（R） is the set of all idempotents of R.

Principal Quasi-Baerness of Rings of Skew Generalized Power Series

Let R be a ring and （S, 〈） be a strictly totally ordered monoid satisfying that 0 〈 s for all s C S. It is shown that if A is a weakly rigid homomorphism, then the skew generalized power series ring [[RS,-〈, λ]] is right p.q.-Baer if and only if R is right p.q.-Baer and any S-indexed subset of S,（R） has a generalized join in S,（R）. Several known results follow as consequences of our results.

Triangular Matrix Representations of Rings of Generalized Power Series

Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either （S,≤） satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].

Morita Duality for the Rings of Generalized Power Series

Let A,B be associative rings with identity,and(S.≤)a strictly totally ordered monoid which is also artinian and finitely generated.For any bimodule AaMB. we show that the bimodule [[A^(S.≤)]][M^(S.≤)][[B^(S.≤)]]defines a Morita duality if and only if _AM_B defines a Morita duality and A is left noetherian.B is right noetherian.As a corollary,it.is shown that the ring[[A^(S.≤)]]of generalized power series over A has a Morita duality if and only if A is a left noetherian ring with a Morita duality induced by a bimodule _AM_B such that B is right noetherian.

PP-Rings of Generalized Power Series

As a generalization of power series rings, Ribenboim introduced the notion of the rings of generalized power series. Let R be a commutative ring, and （S.≤） a strictly totally ordered monoid. We prove that （1） the ring [[R(S.≤]] of generalized power series is a PP-ring if and only if R is a PP-ring and every S-indexed subset C of B（R） （the set of all idempotents of R） has a least upper bound in B（R）. and （2） if （S. ≤） also satisfies the condition that 0≤s for any s∈S, then the ring [[R(S.≤]] is weakly PP if and only if R is weakly PP.

On the P.Q.-Baer Skew Generalized Power Series Modules

For a ring R and a strictly totally ordered monoid(S,≤),letω:S→End(R)be a monoid homomorphism and M an(S,ω)-weakly rigid right R-module(i.e.,for any elements m∈M,b∈R and s∈S,mRb=0 if and only if mω(s)(Rb)=0),where End(R)is the ring of ring endomorphisms of R.It is shown that the skew generalized power series module M[[S]]_(R[[S,ω]])is a principally quasi-Baer module if and only if the annihilator of every submodule generated by an S-indexed subset of M is generated by an idempotent as a right ideal of R.As a consequence we deduce that for an(S,ω)-weakly rigid ring R,the skew generalized power series ring R[[S,ω]]is right principally quasi-Baer if and only if R is right principally quasi-Baer and any S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join in R.The range of previous results in this area is expanded by these results.

Stable Rings for Morita Contexts of Generalized Power Series Rings

In this paper, we show that if rings A and B are （s, 2）-rings, then so is the ring of a Morita Context（[[A^S,≤]],[[B^S,≤]],[[M^S,≤]],[[N^S,≤]],ψ^S,Ф^S）of generalized power series. Also we get analogous results for unit 1-stable ranges, GM-rings and rings which have stable range one. These give new classes of rings satisfying such stable range conditions.

Co-Hopfian Modules of Generalized Inverse Polynomials

Let R be an associative ring not necessarily possessing an identity and （S,≤） a strictly totally ordered monoid which is also artinian and satisfies that 0≤s for any s∈S.Assume that M is a left R-module having property （F）.It is shown that M is a co-Hopfian left R-module if and only if [MS,≤]is a co-Hopfan left [[RS,≤]]-module.

INJECTIVE PRECOVERS AND MODULES OF GENERALIZED INVERSE POLYNOMIALS

This paper is motivated by S. Park [10] in which the injective cover of left R[x]- module M[x? ] of inverse polynomials over a left R-module M was discussed. The 1 author considers the ?-covers of modules and shows that if η : P ?→ M is an ?- cover of M, then [ηS, ] : [PS, ] ?→ [MS, ] is an [?S, ]-cover of left [[RS, ]]-module ≤ ≤ ≤ ≤ ≤ [MS, ], where ? is a class of left R-modules and [MS, ] is the left [[RS, ]]-module of ≤ ≤ ≤ generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[RS, ]]-module [MS, ] are discussed. ≤

Generalized Macaulay-Northcott Modules and Tor-Groups

Let (S,≤) be a strictly totally ordered monoid which is also artinian, and R a right noetherian ring. Assume that M is a finitely generated right R-module and N is a left Rmodule. Denote by [[MS,≤]] and [NS,≤] the module of generalized power series over M, and the generalized Macaulay-Northcott module over N, respectively. Then we show that there exists an isomorphism of Abelian groups:Tori[[ RS,≤]]([[MS,≤]],[NS,≤])≌ s∈S ToriR (M,N).

On Monogeny and Epigeny Classes of Modules

Let (S, ≤) be a strictly totally ordered monoid, and M and N be left R modules. We show the following results: (1) If (S, ≤) is finitely generated and satisfies the condition that 0≤S for any s ∈S, then Epi([[RS,≤]][[MS,≤]]) = Epi([[RS,≤]][[NS,≤]]) if and only if Epi(M) = Epi(N); (2) If (S,≤) is artinian, then Mono([[RS,≤]][MS,≤])= Mono([[RS,≤]][NS,≤]) if and only if Mono(M) = Mono(N).