Fully idempotent near-rings and sheaf representations

Fully idempotent near-rings are defined and characterized which yields information on the lattice of ideals of fully idempotent rings and near-rings. The space of prime ideals is topologized and a sheaf representation is given for a class of fully idempotent near-rings which includes strongly regular near-rings.

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A characterization of minimal prime ideals

Glasgow Mathematical Journal ◽

10.1017/s0017089500032547 ◽

1998 ◽

Vol 40 (2) ◽

pp. 223-236 ◽

Cited By ~ 13

Author(s):

Gary F. Birkenmeier ◽

Jin Yong Kim ◽

Jae Keol Park

Keyword(s):

Positive Integer ◽

Prime Ideal ◽

Commutative Ring ◽

Minimal Prime Ideal ◽

Prime Ideals ◽

Sheaf Representations ◽

Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

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Weakly and Strongly Regular Near-rings

Algebra Colloquium ◽

10.1142/s1005386705000118 ◽

2005 ◽

Vol 12 (01) ◽

pp. 121-130 ◽

Cited By ~ 1

Author(s):

N. Argac ◽

N. J. Groenewald

Keyword(s):

Affirmative Answer ◽

Prime Ideals ◽

Basic Properties ◽

Strongly Regular

In this paper, we prove some basic properties of left weakly regular near-rings. We give an affirmative answer to the question whether a left weakly regular near-ring with left unity and satisfying the IFP is also right weakly regular. In the last section, we use among others left 0-prime and left completely prime ideals to characterize strongly regular near-rings.

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Prime ideals and sheaf representation of a pseudo symmetric ring

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1973-0338058-9 ◽

1973 ◽

Vol 184 ◽

pp. 43-43 ◽

Cited By ~ 116

Author(s):

Gooyong Shin

Keyword(s):

Prime Ideals ◽

Sheaf Representation

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Almost-Dedekind rings

Glasgow Mathematical Journal ◽

10.1017/s0017089500030640 ◽

1994 ◽

Vol 36 (1) ◽

pp. 131-134 ◽

Cited By ~ 1

Author(s):

E. W. Johnson

Keyword(s):

Commutative Ring ◽

Integral Domain ◽

Classical Result ◽

Dedekind Domain ◽

Ring Theory ◽

Prime Ideals ◽

Principal Ideals ◽

Commutative Ring Theory ◽

Lattice Of Ideals

Throughout we assume all rings are commutative with identity. We denote the lattice of ideals of a ringRbyL(R), and we denote byL(R)* the subposetL(R)−R.A classical result of commutative ring theory is the characterization of a Dedekind domain as an integral domainRin which every element ofL(R)* is a product of prime ideals (see Mori [5] for a history). This result has been generalized in a number of ways. In particular, rings which are not necessarily domains but which otherwise satisfy the hypotheses (i.e. general ZPI-rings) have been widely studied (see, for example, Gilmer [3]), as have rings in which only the principal ideals are assumed to satisfy the hypothesis (i.e. π-rings).

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On a sheaf representation of a class of near-rings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700017365 ◽

1977 ◽

Vol 23 (1) ◽

pp. 78-83 ◽

Cited By ~ 5

Author(s):

George Szeto

Keyword(s):

Prime Ideals ◽

Zariski Topology ◽

Sheaf Representation

AbstractIt is proved that R is a near-ring with identity in which every element is a power of itself if and only if it is isomorphic with a near-ring of sections of a sheaf of near-fields in which every element is a power of itself. We also obtain that the Boolean spectrum is homeomorphic with the space of all completely prime ideals of R with the Zariski topology.

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N-ideals of commutative rings

Filomat ◽

10.2298/fil1710933t ◽

2017 ◽

Vol 31 (10) ◽

pp. 2933-2941 ◽

Cited By ~ 3

Author(s):

Unsal Tekir ◽

Suat Koc ◽

Kursat Oral

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Proper Ideal ◽

Prime Ideals ◽

Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

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