Commutative Rings with a Prescribed Number of Isomorphism Classes of Minimal Ring Extensions

2017 ◽  
pp. 145-158 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Commutative Rings ◽  
Isomorphism Classes ◽  
Ring Extensions

Commutative rings and modules that are Nil*-coherent or special Nil*-coherent

2017 ◽  
Vol 16 (10) ◽  
pp. 1750187 ◽  
Author(s):  
Karima Alaoui Ismaili ◽  
David E. Dobbs ◽  
Najib Mahdou
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Basic Properties ◽  
Unital Ring ◽  
Reduced Ring ◽  
Coherent Ring ◽  
Ring Extensions ◽  
Coherent Rings ◽  
Finitely Presented

Recently, Xiang and Ouyang defined a (commutative unital) ring [Formula: see text] to be Nil[Formula: see text]-coherent if each finitely generated ideal of [Formula: see text] that is contained in Nil[Formula: see text] is a finitely presented [Formula: see text]-module. We define and study Nil[Formula: see text]-coherent modules and special Nil[Formula: see text]-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil[Formula: see text]-coherent ring and any special Nil[Formula: see text]-coherent ring is a Nil[Formula: see text]-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil[Formula: see text]-coherent ring (regardless of whether it is coherent). Several examples of Nil[Formula: see text]-coherent rings that are not special Nil[Formula: see text]-coherent rings are obtained as byproducts of our study of the transfer of the Nil[Formula: see text]-coherent and the special Nil[Formula: see text]-coherent properties to trivial ring extensions and amalgamated algebras.

Almost Bézout rings and almost GCD-rings

2019 ◽  
Vol 13 (06) ◽  
pp. 2050107
Author(s):  
Abdelhaq El Khalfi ◽  
Najib Mahdou
Keyword(s):  
Commutative Rings ◽  
Ring Extensions

In this paper, we study the possible transfer of the property of being an [Formula: see text]-ring to trivial ring extensions and amalgamated algebras along an ideal. Also, we extend the notion of an almost GCD-domain to the context of arbitrary rings, and we study the possible transfer of this notion to trivial ring extensions and amalgamated algebras along an ideal. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

Finite commutative rings with higher genus unit graphs

2014 ◽  
Vol 14 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Huadong Su ◽  
Kenta Noguchi ◽  
Yiqiang Zhou
Keyword(s):  
Nonnegative Integer ◽  
Commutative Rings ◽  
Orientable Surface ◽  
Simple Graph ◽  
Isomorphism Classes ◽  
Unit Graph ◽  
Higher Genus ◽  
Vertex Set ◽  
Finite Commutative Rings

Let R be a ring with identity. The unit graph of R, denoted by G(R), is a simple graph with vertex set R, and where two distinct vertices x and y are adjacent if and only if x + y is a unit in R. The genus of a simple graph G is the smallest nonnegative integer g such that G can be embedded into an orientable surface Sg. In this paper, we determine all isomorphism classes of finite commutative rings whose unit graphs have genus at most three.

Nilpotent graphs of genus one

2014 ◽  
Vol 06 (03) ◽  
pp. 1450037
Author(s):  
R. Kala ◽  
S. Kavitha
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Isomorphism Classes ◽  
Vertex Set ◽  
Genus One ◽  
Finite Commutative Rings

Let R be a commutative ring with identity. The nilpotent graph of R, denoted by ΓN(R), is a graph with vertex set [Formula: see text], and two vertices x and y are adjacent if and only if xy is nilpotent, where [Formula: see text] is nilpotent, for some y ∈ R*}. In this paper, we determine all isomorphism classes of finite commutative rings with identity whose ΓN(R) has genus one.

The genus two class of graphs arising from rings

2018 ◽  
Vol 17 (10) ◽  
pp. 1850193 ◽  
Author(s):  
T. Asir
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Commutative Rings ◽  
Isomorphism Classes ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Genus Two

Given a commutative ring [Formula: see text] with identity [Formula: see text], its Jacobson graph [Formula: see text] is defined to be the graph in which the vertex set is [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Here [Formula: see text] denotes the Jacobson radical of [Formula: see text] and [Formula: see text] is the set of unit elements in [Formula: see text]. This paper investigates the genus properties of Jacobson graph. In particular, we determine all isomorphism classes of commutative rings whose Jacobson graph has genus two.

Rings in which every regular locally principal ideal is projective

2019 ◽  
Vol 56 (2) ◽  
pp. 241-251
Author(s):  
Rachida El Khalfaoui ◽  
Najib Mahdou
Keyword(s):  
Principal Ideal ◽  
Commutative Rings ◽  
Direct Products ◽  
Ring Extensions

Abstract In this article, we study the class of rings in which every regular locally principal ideal is projective called LPP-rings. We investigate the transfer of this property to various constructions such as direct products, amalgamation of rings, and trivial ring extensions. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned property.

On PM-rings, rings of finite character and h-local rings

2014 ◽  
Vol 13 (06) ◽  
pp. 1450018
Author(s):  
N. Mahdou ◽  
A. Mimouni ◽  
M. A. S. Moutui
Keyword(s):  
Commutative Rings ◽  
Local Rings ◽  
Local Properties ◽  
Ring Extensions

In this paper, we investigate the transfer of the notions of pm-rings, rings of finite character and h-local rings to trivial ring extensions of rings by modules, amalgamations of rings along ideals and pullbacks. Our aim is to provide new classes of commutative rings satisfying these properties and our results generate new families of examples of rings for which the finite character and semi-local properties are equivalents.

scholarly journals Epimorphic flat maps

1986 ◽  
Vol 29 (1) ◽  
pp. 57-59 ◽  
Author(s):  
Frederick W. Call
Keyword(s):  
Commutative Rings ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Condition ◽  
Closed Set ◽  
Canonical Sheaf ◽  
Isomorphism Classes ◽  
Geometric Theorem ◽  
Necessary And Sufficient ◽  
Geometric Formulation ◽  
Affine Scheme

In this note, we derive a necessary and sufficient condition for a flat map of (commutative) rings to be a flat epimorphism. Flat epimorphisms φ:A → B(i.e.φ is an epimorphism in the category of rings, and the ring B is flat as an A-module) have been studied by several authors in different forms. Flat epimorphisms generalize many of the results that hold for localizations with respect to a multiplicatively closed set (see, for example [6]).In a geometric formulation, D. Lazard [3, Chapitre IV, Proposition 2.5] has shown that isomorphism classes of flat epimorphisms from a ring A are in 1-1 correspondence with those subsets of Spec A such that the sheaf structure induced from the canonical sheaf structure of Spec A yields an affine scheme. N. Popescu and T. Spircu [4, Théorème 2.7] have given a characterization for a ring homomorphism to be a flat epimorphism, but our characterization, under the assumption of flatness is easier to apply. For corollaries, we can obtain known results due to D. Lazard, T. Akiba, and M. F. Jones, and generalize a geometric theorem of D. Ferrand.

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