<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The main object of this paper is to study relative homological aspects as well as further properties of <italic>τ</italic> -closed submodules. A submodule <italic>N</italic> of a module <italic>M</italic> is said to be <italic>τ</italic> -closed (or <italic>τ</italic> -pure) provided that <italic>M/N</italic> is <italic>τ</italic> -torsion-free, where <italic>τ</italic> stands for an idempotent radical. Whereas the well-known proper class 𝒞<italic>losed</italic> (𝒫<italic>ure</italic>) of closed (pure) short exact sequences, the class <italic>τ −</italic>𝒞<italic>losed</italic> of <italic>τ</italic> -closed short exact sequences need not be a proper class. We describe the smallest proper class <italic>〈τ −</italic> 𝒞<italic>losed〉</italic> containing <italic>τ −</italic> 𝒞<italic>losed</italic>, through <italic>τ</italic> -closed submodules. We show that the smallest proper class <italic>〈τ −</italic> 𝒞<italic>losed〉</italic> is the proper classes projectively generated by the class of <italic>τ</italic> -torsion modules and coprojectively generated by the class of <italic>τ</italic> -torsion-free modules. Also, we consider the relations between the proper class <italic>〈τ −</italic> 𝒞<italic>losed〉</italic> and some of well-known proper classes, such as 𝒞<italic>losed,</italic> 𝒫<italic>ure</italic>.</p></abstract>

The main object of this paper is to study relative homological aspects as well as further properties of τ -closed submodules. A submodule N of a module M is said to be τ -closed (or τ -pure) provided that M/N is τ -torsion-free, where τ stands for an idempotent radical. Whereas the well-known proper class 𝒞losed (𝒫ure) of closed (pure) short exact sequences, the class τ −𝒞losed of τ -closed short exact sequences need not be a proper class. We describe the smallest proper class 〈τ − 𝒞losed〉 containing τ − 𝒞losed, through τ -closed submodules. We show that the smallest proper class 〈τ − 𝒞losed〉 is the proper classes projectively generated by the class of τ -torsion modules and coprojectively generated by the class of τ -torsion-free modules. Also, we consider the relations between the proper class 〈τ − 𝒞losed〉 and some of well-known proper classes, such as 𝒞losed, 𝒫ure.

Proper classes generated by <italic>τ</italic>- closed submodules

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

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Proper classes generated by - closed submodules

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The main object of this paper is to study relative homological aspects as well as further properties of τ -closed submodules. A submodule N of a module M is...

Proper classes generated by τ- closed submodules

Published Online: Dec 21, 2019

Page range: 83 - 95

Received: Jan 03, 2019

Accepted: Jan 31, 2019

DOI: https://doi.org/10.2478/auom-2019-0035

Keywords
(-)closed submodule, proper class, pure submodule, Goldie’s torsion theory, Dickson’s torsion theory

© 2019 Yılmaz Durğun et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Proper classes generated by τ- closed submodules

Published Online: Dec 21, 2019

Page range: 83 - 95

Received: Jan 03, 2019

Accepted: Jan 31, 2019

DOI: https://doi.org/10.2478/auom-2019-0035

Keywords(-)closed submodule, proper class, pure submodule, Goldie’s torsion theory, Dickson’s torsion theory

© 2019 Yılmaz Durğun et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
(-)closed submodule, proper class, pure submodule, Goldie’s torsion theory, Dickson’s torsion theory