In this article, we give several new characterizations of Quasi-Frobenius rings by using mininjectivity, simple injectivity, and small injectivity, respectively. Several known results on Quasi-Frobenius rings are reproved as corollaries.

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The Faith-Menal conjecture is one of the three main open conjectures on QF rings. It says that every right noetherian and left FP -injective ring is QF. In this paper, it is proved that the conjecture is true if every nonzero complement left ideal of the ring R is not small (or not singular).Several known results are then obtained as corollaries.

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A ring R is called a J-regular ring if R/J(R) is von Neumann regular, where J(R) is the Jacobson radical of R. It is proved that if R is Jregular, then (i) R is right n-injective if and only if every homomorphism from an n-generated small right ideal of R to R R can be extended to one fromSome known results are improved.Proposition 4. If R is J-regular, then every factor ring S of R is also Jregular.Proof. Let S be a factor ring of R and φ be the ring epimorphism from R to S. By [1, Corollary 15.8The following two propositions show that being J-regular is a Morita invariant.Proposition 6. If R is J-regular, then eRe is also J-regular, where e 2 = e ∈ R.Proof. We only need to show that for each a ∈ eRe, there exist b ∈ eRe and c ∈ J(eRe) = eJe (see [2, Theorem 21.10]) such that a = aba + c. As R is J-regular, there exist b ′ ∈ R and c ′ ∈ J such that a = ab ′ a + c ′ . Since a ∈ eRe, a = ab ′ a + c ′ = aeb ′ ea + c ′ . It is clear that c ′ = a − ab ′ a ∈ eRe ∩ J = eJe. Then we can set b = eb ′ e and c = c ′ . Proposition 7. If R is J-regular, then every matrix ring M n×n (R) is also J-regular, n ≥ 1.Proof. It is well-known that J(M n×n (R)) = M n×n (J) (see [2, Page 61]). And it is also easy to prove that M n×n (R)Theorem 8. Let R be a J-regular ring and K a finitely generated projective right R-module. Then the endomorphism ring End (K) of K is also J-regular.Proof. Since K is finitely generated and projective, K is a direct summand of a finitely generated free right R-module F . Then there exists some integer n ≥ 1