On minimal strongly prime ideals

In this paper we give some characterizations of a ring Rwhose unique maximal nil ideal N _r (R) coincides with the set of all its nilpotent elements N(R) by using its minimal strongly prime ideals.     

Pseudocomplementation and minimal prime ideals in semirings

In Section 2 of the present paper, we introduce the concept of pseudocomplementation for semirings and show the semiring version of some known results in lattice theory. We also introduce semirings with pc-functions and prove some interesting results for minimal prime ideals of such semirings. In Section 3, some classical results for minimal prime ideals in ring theory are generalized in the context of semiring theory.     

Maximal and minimal prime ideals of incidence algebras

Abstract

In this paper we study the relationship between some homological properties such as the weak and the global dimension, the strong n-coherence, and the (n, d)-property, where n and d are two integers, of a commutative ring and its subrings retract. A special application is the transfer of these properties from a commutative ring to its fixed subring with respect to a subgroup of its group of automorphisms. It concludes with a discussion of the scopes and limits of our results.     

On minimal prime ideals of commutative Bezout rings

We study the spectrum of minimal prime ideals of commutative Bezout rings. We apply the results obtained to the problem of diagonal reduction of matrices over rings of this sort. Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 7, pp. 1001–1005, July, 1999.     

Invariance of minimal prime ideals of nonassociative rings under multiplicative isomorphisms

It is proved that the image of a minimal prime ideal under a multiplicative isomorphism in a nonassociative ring is also a minimal prime ideal and congruence with respect to its modulus is preserved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 90–92, 1989.     

Space of minimal prime ideals of a ring without nilpotent elements

Hull-kernel topology on the set ∑(R) of prime ideals of a ring R with unity and without nilpotent elements is discussed. The restriction of this topology to the set π(R) of minimal prime ideals of R has been investigated in detail. The compactness of π(R) has been characterized in several ways. An interesting characterization of Baer rings is given.A functorial correspondence between the category of rings having the property that every prime ideal contains a unique minimal prime ideal and their minimal spectra is established.     

The space of minimal prime ideals in a 0-distributive semilattice

LetS be a 0-distributive semilattice and be its minimal spectrum. It is shown that is Hausdorff. The compactness of has been characterized in several ways. A representation theorem (like Stone's theorem for Boolean algebras) for disjunctive, 0-distributive semilattices is obtained.     

Unions of minimal prime ideals in rings of continuous functions on compact spaces

As is well-known to those familiar with the Gillman-Jerison text on the ring C(X) of real-valued continuous functions on a topological space (which we may assume is a Tychonoff space), every prime ideal is contained in a unique maximal ideal. In case X is compact, each maximal ideal is of the form M _p for some ${p \in X}$ and consists of all ${f \in C(X)}$ such that f(p) = 0, while the intersection of all minimal prime ideals contained in M _p is the set of all continuous functions which vanish on a neighborhood of p. In this paper, we reverse some inclusions and study the union of all of the minimal prime ideals contained in M _p ; particularly in the case when this set-theoretic union is all of M _p . When this occurs, we call X a UMP-space. By making use of the well-known theorem of Gelfand and Kolmogoroff, we obtain new results without assuming that X is compact. It turns out that all UMP-spaces have the property that each of its nonempty zero-sets has nonempty interior. That is, X is an almost P-space. But this condition is far from sufficient.     

Regularity of prime ideals

Mathematische Zeitschrift - We answer several natural questions which arise from a recent paper of McCullough and Peeva providing counterexamples to the Eisenbud–Goto Regularity Conjecture....     

Estimates for prime ideals

A modification of an old argument due to ?ebi?ev is used to obtain uniform estimates for prime ideals in algebraic number fields.