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- Title
Purity and approximation with respect to a class of morphisms.
- Authors
Mao, Lixin
- Abstract
We investigate purity and approximation with respect to a class of morphisms of modules. Let ℐ be a class of left R -module morphisms. An epimorphism ϕ : A → B of left R -modules is called ℐ -pure if for any morphism ψ : X → B in ℐ , there is a morphism : X → A such that ϕ = ψ. An ℐ -pure epimorphism f : M → N is called ℐ -superfluous if every morphism g : A → M with f g ℐ -pure is itself ℐ -pure. We get many properties of ℐ -pure and ℐ -superfluous morphisms. As an application, we generalize the classical characterization of a projective cover to a general setting of an ℐ -cover. It is proven that an epimorphism φ : M → N in ℐ is an ℐ -cover of N if and only if N has an ℐ -cover and φ is an ℐ -superfluous epimorphism if and only if φ is ℐ -pure and there is no proper submodule K of M such that ker (φ) + K = M and φ | K is ℐ -pure. In addition, some dual results are also given.
- Subjects
RHYTHM &; blues music; MORPHISMS (Mathematics)
- Publication
Journal of Algebra & Its Applications, 2021, Vol 20, Issue 9, p1
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S021949882150170X