JournalsrsmupVol. 144pp. 27–43

P1\mathcal P_1-covers over commutative rings

  • Silvana Bazzoni

    Università degli Studi di Padova, Italy
  • Giovanna Le Gros

    Università degli Studi di Padova, Italy
$\mathcal P_1$-covers over commutative rings cover

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In this paper we consider the class P1(R)\mathcal P_1(R) of modules of projective dimension at most one over a commutative ring RR and we investigate when P1(R)\mathcal P_1(R) is a covering class. More precisely, we investigate Enochs' Conjecture, that is the question of whether P1(R)\mathcal P_1(R) is covering necessarily implies that P1(R)\mathcal P_1(R) is closed under direct limits. We answer the question affirmatively in the case of a commutative semihereditary ring RR. This gives an example of a cotorsion pair (P1(R),P1(R))(\mathcal P_1(R), \mathcal P_1(R)^\perp) which is not necessarily of finite type such that P1(R)\mathcal P_1(R) satisfies Enochs' Conjecture. Moreover, we describe the class limP1(R)\varinjlim \mathcal P_1(R) over (not necessarily commutative) rings which admit a classical ring of quotients.

Cite this article

Silvana Bazzoni, Giovanna Le Gros, P1\mathcal P_1-covers over commutative rings. Rend. Sem. Mat. Univ. Padova 144 (2020), pp. 27–43

DOI 10.4171/RSMUP/54