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