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 genera...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.展开更多
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),w...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.展开更多
基金The NNSF (10171082) of China and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.C.
文摘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.
文摘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.
