JournalsrsmupVol. 129pp. 115–127

The Amalgamated Duplication of a Ring Along a Semidualizing Ideal

  • Maryam Salimi

    Islamic Azad University, Teheran, Iran
  • Elham Tavasoli

    Islamic Azad University, Teheran, Iran
  • S. Yassemi

    Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
The Amalgamated Duplication of a Ring Along a Semidualizing Ideal cover
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Abstract

Let RR be a commutative Noetherian ring and let II be an ideal of RR. In this paper, after recalling briefly the main properties of the amalgamated duplication ring RIR\bowtie I which is introduced by D'Anna and Fontana, we restrict our attention to the study of the properties of RIR\bowtie I, when II is a semidualizing ideal of RR, i.e., II is an ideal of RR and II is a semidualizing RR-module. In particular, it is shown that if II is a semidualizing ideal and MM is a finitely generated RR-module, then MM is totally II-reflexive as an RR-module if and only if MM is totally reflexive as an (RI)(R\bowtie I)-module. In addition, it is shown that if II is a semidualizing ideal, then RR and II are Gorenstein projective over RIR \bowtie I, and every injective RR-module is Gorenstein injective as an (RI)(R \bowtie I)-module. Finally, it is proved that if II is a non-zero flat ideal of RR, then fd R(M)=fdRI(MR(RI))=fdR(MR(RI))_{R}(M) = {\rm fd} _{R \bowtie I} (M \otimes _{R} (R\bowtie I)) = {\rm fd} _{R} (M\otimes _{R}(R\bowtie I)), for every RR-module MM.

Cite this article

Maryam Salimi, Elham Tavasoli, S. Yassemi, The Amalgamated Duplication of a Ring Along a Semidualizing Ideal. Rend. Sem. Mat. Univ. Padova 129 (2013), pp. 115–127

DOI 10.4171/RSMUP/129-8