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Covers and Envelopes by Submodules or Quotient-Modules

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Let R be a ring, X a class of left R -modules, S the class of submodules of X , and Q the class of quotient-modules of X . It… Click to show full abstract

Let R be a ring, X a class of left R -modules, S the class of submodules of X , and Q the class of quotient-modules of X . It is shown that S Q is precovering (preenveloping) if and only if every injective (projective) left R -module has an X -precover ( X -preenvelope). Both epic and monic S -(pre) covers ( Q -(pre) envelopes) are studied. Moreover, some applications are given. In particular, it is proven that the injective envelope of any projective left R -module is projective if and only if the class of quotient-modules of projective and injective left R -modules is monic preenveloping.

Keywords: jats inline; xmlns http; inline formula; formula math; math xmlns; math

Journal Title: Journal of Mathematics
Year Published: 2023

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