An alternative perspective on pure-projectivity of modules
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An alternative perspective on pure‑projectivity of modules Yusuf Alagöz1 · Yılmaz Durg̃un2 Accepted: 23 September 2020 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract The study of pure-projectivity is accessed from an alternative point of view. Given modules M and N, M is said to be N-pure-subprojective if for every pure epimorphism g ∶ B → N and homomorphism f ∶ M → N , there exists a homomorphism h ∶ M → B such that gh = f . For a module M, the pure-subprojectivity domain of M is defined to be the collection of all modules N such that M is N-pure-subprojective. We obtain characterizations for various types of rings and modules, including FPinjective and FP-projective modules, von Neumann regular rings and pure-semisimple rings in terms of pure-subprojectivity domains. As pure-subprojectivity domains clearly include all pure-projective modules, a reasonable opposite to pure-projectivity in this context is obtained by considering modules whose pure-subprojectivity domain consists of only pure-projective. We refer to these modules as psp-poor. Properties of pure-subprojectivity domains and of psp-poor modules are studied. Keywords Pure-projective modules · Pure-subprojectivity domains · psp-poor modules · Pure-semisimple rings Mathematics Subject Classfication 16D40 · 16D80 · 18G25
1 Introduction and preliminaries Throughout, R will denote an associative ring with identity, and modules will be unital right R-modules, unless otherwise stated. As usual, we denote by Mod−R the category of right R-modules. For a module M, E(M), PE(M), and M + denote the injective hull, the pure-injective preenvelope and the character module Homℤ (M, ℚ∕ℤ) , respectively. Communicated by Sergio R. López-Permouth. * Yusuf Alagöz [email protected] 1
Department of Mathematics, Siirt University, Siirt, Turkey
2
Department of Mathematics, Çukurova University, Adana, Turkey
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Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
Some new studies in module theory have focused on to approach to the projectivity from the point of relative notions. For a module M, the projectivity domain of M, 𝔓𝔯−1 (M) , is defined to be the collection of all modules N such that M is projective relative to N (or N-projective) [3]. Holston et al. in [11] initiated the study of projectively poor (p-poor) modules, namely modules whose projectivity domains consist only of semisimple modules in Mod−R , and rings with no p-middle class. The study of rings with no middle class has a growing interest in recent years (see [4, 5, 15]). In contrast to the notion of relative projectivity, Holston et al. introduced in [12] the notion of subprojectivity. Namely, a module M is said to be N-subprojective if for every epimorphism g ∶ B → N and homomorphism f ∶ M → N , then there exists a homomorphism h ∶ M → B such that gh = f . For a module M, the subprojectivity domain of M, 𝔓𝔯−1 (M) , is defined to be the collection of all modules N such that M is N-subprojective, that is 𝔓𝔯−1 (M) = {N : M is N-subprojecti
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