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- W1645398439 abstract "The Faith-Menal conjecture says that every strongly right $Johns$ ring is $QF$. The conjecture is also equivalent to say every right noetherian left $FP$-injective ring is $QF$. In this short article, we show that the conjecture is true under the condition(a proper generalization of left $CS$ condition)that every nonzero complement left ideal is not small(a left ideal $I$ is called small if for every left ideal $K$, $K$+$I$=$R$ implies $K$=$R$). It is also proved that (1) $R$ is $QF$ if and only if $R$ is a left and right mininjective ring with $ACC$ on right annihilators in which $S_{r}subseteq ^{ess}R_{R}$; (2) $R$ is $QF$ if and only if $R$ is a right simple injective ring with $ACC$ on right annihilators in which $S_{r}subseteq ^{ess}R_{R}$. Several known results on $QF$ rings are obtained as corollaries." @default.
- W1645398439 created "2016-06-24" @default.
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- W1645398439 date "2005-04-05" @default.
- W1645398439 modified "2023-09-27" @default.
- W1645398439 title "A Note on Quasi-Frobenius Rings" @default.
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