On the quotient ring of commutative rings with \(acc^{\perp}\) on annihilator ideals
There is not much in this short paper that is not known. Let R be a commutative ring which satisfies ACC on annihilators. Then R has only a finite number of maximal annihilators, which are prime, and every ideal consisting of zero divisors has non-zero annihilator [see \textit{I. Kaplansky}, ``Commutative rings'' (1970; Zbl 0203.346); theorem 80]. Let N denote the prime radical of R. It is proved that the classical quotient ring Q of R is Artinian if and only if every element of R regular modulo N is regular in R. [This result can be found in a paper by \textit{C. R. Hajarnavis}, J. Lond. Math. Soc., II. Ser. 5, 596-600 (1972; Zbl 0244.16003).] The author also discusses when Q is QF, equivalently FPF.
scientific article; zbMATH DE number 4204518