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It is known that a ring is left Noetherian if and only if every left -module has an injective (pre)cover. We show that if is a right -coherent ring, then every right -module has an -injective (pre)cover; if is a ring such that every -injective right -module is -pure extending, and if every right -module has an -injective cover, then is right -coherent. As applications of these results, we give some characterizations of -rings, von Neumann regular rings and semisimple rings....
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.
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