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.
Zelmanowitz [12] introduced the concept of ring, which we call right zip rings, with the defining properties below, which are equivalent:(ZIP 1) If the right anihilator X⊥ of a subset X of R is zero, then X1⊥ = 0 for a finite subset X1 ⊆ X.(ZIP 2) If L is a left ideal and if L⊥ = 0, then L1⊥ = 0 for a finitely generated left ideal L1 ⊆ L.In [12], Zelmanowitz noted that any ring R satisfying the d.c.c. on anihilator right ideals (= dcc ⊥) is a right zip ring, and hence, so is any subring of R. He...
We give some new characterizations of quasi-Frobenius rings by the generalized injectivity of rings. Some characterizations give affirmative answers to some open questions about quasi-Frobenius rings; and some characterizations improve some results on quasi-Frobenius rings.