The minimal prime spectrum of rings with annihilator conditions.
A ring \(R\) is said to have the right annihilator condition (a.c.) if the right annihilator of any finitely generated ideal of \(R\) equals the right annihilator of some principal ideal. Many rings are shown to have the (right and left) (a.c.), such as semiprime rings with finitely many minimal prime ideals; biregular rings; skew polynomial rings (with an automorphism) over semiprime rings. Further, (a.c.) is preserved in matrix rings and in classical quotient rings. The minimal prime spectrum of \(R\) is the set \(\text{Min}(R)\) of minimal prime ideals of \(R\), equipped with the relative hull-kernel (Zariski) topology from \(\text{Spec}(R)\). The authors prove that if \(R\) is a reduced ring with classical right quotient ring \(Q(R)\), the following conditions are equivalent: (1) \(Q(R)\) is strongly regular; (2) \(\text{Min}(R)\) is compact and \(R\) has (a.c.); (3) \(\text{Min}(R)\) is compact and every finitely generated ideal of \(R\) consisting of left (right) zero-divisors has nonzero right (left) annihilator. This generalizes results in the commutative case due to \textit{M. Henriksen} and \textit{M. Jerison} [Trans. Am. Math. Soc. 115, 110-130 (1965; Zbl 0147.29105)], \textit{Y. Quentel} [Bull. Soc. Math. Fr. 99, 265-272 (1971); erratum ibid. 100, 461 (1972; Zbl 0215.36803)], and \textit{J. A. Huckaba} and \textit{J. M. Keller} [Pac. J. Math. 83, 375-379 (1979; Zbl 0388.13001)].
scientific article; zbMATH DE number 5544397
scientific article