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Let be a ring and a right -module. is called -cofinitely supplemented if every submodule of with finitely generated has a supplement that is a direct summand of . In this paper various properties of the -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of -cofinitely supplemented modules is -cofinitely supplemented. (2) A ring is semiperfect if and only if every free -module is -cofinitely supplemented. In addition, if has the summand sum...
The duals of -compact modules are briefly discussed.
Let0 → ∏ℵI Mα ⎯λ→ ∏I Mα ⎯γ→ Coker λ → 0 be an exact sequence of modules, in which ℵ is an infinite cardinal, λ the natural injection and γ the natural surjection. In this paper, the conditions are given mainly in the four theorems so that λ (γ respectively) is split or locally split. Consequently, some known results are generalized. In particular, Theorem 1 of [7] and Theorem 1.6 of [5] are improved.
Given a hereditary torsion theory in Mod-, a module is called -supplemented if every submodule of contains a direct summand of with torsion. A submodule of is called -supplement of in if and and is -weakly supplemented if every submodule...
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