scholarly journals Primitiewe elemente vir kommutatiewe ringuitbreidings

1991 ◽  
Vol 10 (2) ◽  
pp. 67-71
Author(s):  
H. J. Schutte
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Integral Domain ◽  
Prime Ideals ◽  
Primitive Elements ◽  
Integral Domains ◽  
Field Extensions ◽  
Ring Extensions ◽  
Minimal Prime

The existence of primitive elements for integral domain extensions is considered with reference to the well known theorem about primitive elements for field extensions. Primitive elements for extensions of a commutative ring R with identity are considered, where R has only a finite number of minimal prime ideals with zero intersection. This case is reduced to the case for ring extensions of integral domains.

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
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.

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

scholarly journals A Remark on Coherent Overrings

1978 ◽  
Vol 21 (3) ◽  
pp. 373-375 ◽  
Author(s):  
Ira J. Papick
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
General Theorem ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Quotient Field ◽  
Valuation Rings ◽  
Finitely Presented

Throughout this note, let R be a (commutative integral) domain with quotient field K. A domain S satisfying R ⊆ S ⊆ K is called an overring of R, and by dimension of a ring we mean Krull dimension. Recall [1] that a commutative ring is said to be coherent if each finitely generated ideal is finitely presented.In [2], as a corollary of a more general theorem, Davis showed that if each overring of a domain R is Noetherian, then the dimension of R is at most 1. (This corollary is the converse of a version of the Krull-Akizuki Theorem [5, Theorem 93], and can also be proved directly by using the existence of valuation rings dominating finite chains of prime ideals [4, Corollary 16.6].) It is our purpose to prove that if R is Noetherian and each overring of R is coherent, then the dimension of £ is at most 1. We shall also indicate some related questions and examples.

Some Results on Annihilating-ideal Graphs

2016 ◽  
Vol 59 (3) ◽  
pp. 641-651
Author(s):  
Farzad Shaveisi
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Bipartite Graph ◽  
Maximum Degree ◽  
Complete Bipartite Graph ◽  
Independence Number ◽  
Prime Ideals ◽  
Reduced Ring ◽  
Vertex Set ◽  
Minimal Prime

AbstractThe annihilating-ideal graph of a commutative ring R, denoted by 𝔸𝔾(R), is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices I and J are adjacent if and only if IJ = (0). Here we show that if R is a reduced ring and the independence number of 𝔸𝔾(R) is finite, then the edge chromatic number of 𝔸𝔾(R) equals its maximum degree and this number equals 2|Min(R)|−1 also, it is proved that the independence number of 𝔸𝔾(R) equals 2|Min(R)|−1, where Min(R) denotes the set of minimal prime ideals of R. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a ûnite graph 𝔸𝔾(R) is not Eulerian, and that it is Hamiltonian if and only if R contains no Gorenstain ring as its direct summand.

scholarly journals On irreducible divisor graphs in commutative rings with zero-divisors

2015 ◽  
Vol 46 (4) ◽  
pp. 365-388
Author(s):  
Christopher Park Mooney
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Integral Domains ◽  
Graph Theoretic ◽  
Zero Divisors ◽  
Classical Paper

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in integral domains. This inspired the so called irreducible divisor graph of an integral domain studied by J. Coykendall and J. Maney. Factorization in rings with zero-divisors is considerably more complicated than integral domains and has been widely studied recently. We find that many of the same techniques can be extended to rings with zero-divisors. In this article, we construct several distinct irreducible divisor graphs of a commutative ring with zero-divisors. This allows us to use graph theoretic properties to help characterize finite factorization properties of commutative rings, and conversely.

Primary Ideals and Prime Power Ideals

1966 ◽  
Vol 18 ◽  
pp. 1183-1195 ◽  
Author(s):  
H. S. Butts ◽  
Robert W. Gilmer
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Prime Power ◽  
Ideal Theory ◽  
Dedekind Domain ◽  
Primary Ideal ◽  
Primary Ideals ◽  
The Ideal

This paper is concerned with the ideal theory of a commutative ringR.We sayRhas Property (α) if each primary ideal inRis a power of its (prime) radical;Ris said to have Property (δ) provided every ideal inRis an intersection of a finite number of prime power ideals. In (2, Theorem 8, p. 33) it is shown that ifDis a Noetherian integral domain with identity and if there are no ideals properly between any maximal ideal and its square, thenDis a Dedekind domain. It follows from this that ifDhas Property (α) and is Noetherian (in which caseDhas Property (δ)), thenDis Dedekind.

CPI-Extensions: Overrings of Integral Domains with Special Prime Spectrums

1977 ◽  
Vol 29 (4) ◽  
pp. 722-737 ◽  
Author(s):  
Monte B. Boisen ◽  
Philip B. Sheldon
Keyword(s):  
Topological Space ◽  
Partially Ordered Set ◽  
Integral Domain ◽  
Ordered Set ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Integral Domains ◽  
Partially Ordered ◽  
Set Inclusion

Throughout this paper the term ring will denote a commutative ring with unity and the term integral domain will denote a ring having no nonzero divisors of zero. The set of all prime ideals of a ring R can be viewed as a topological space, called the prime spectrum of R, and abbreviated Spec (R), where the topology used is the Zariski topology [1, Definition 4, § 4.3, p. 99]. The set of all prime ideals of R can also be viewed simply as aposet - that is, a partially ordered set - with respect to set inclusion. We will use the phrase the pospec of R, or just Pospec (/v), to refer to this partially ordered set.

On the associativity of the torsion functor

1974 ◽  
Vol 10 (1) ◽  
pp. 107-118
Author(s):  
John Clark
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Noetherian Ring ◽  
Integral Domain ◽  
Direct Sum ◽  
Torsion Functor

Let R be a commutative ring with identity. We say that tor is associative over R if for all R-modules A, B, C there is an isomorphism Our main results are that (1) tor is associative over a noetherian ring R if and only if R is the direct sum of a finite number of Dedekind rings and uniserial rings, and (2) tor is associative over an integral domain R if and only if R is a Prüfer ring.

Pairs of domains where most of the intermediate domains are Prüfer

2020 ◽  
pp. 2150101
Author(s):  
Noômen jarboui
Keyword(s):  
Integral Domain ◽  
Prüfer Domain ◽  
Integral Domains ◽  
Integrally Closed ◽  
Ring Extensions

Let [Formula: see text] be an extension of integral domains. The ring [Formula: see text] is said to be maximal non-Prüfer subring of [Formula: see text] if [Formula: see text] is not a Prüfer domain, while each subring of [Formula: see text] properly containing [Formula: see text] is a Prüfer domain. Jaballah has characterized this kind of ring extensions in case [Formula: see text] is a field [A. Jaballah, Maximal non-Prüfer and maximal non-integrally closed subrings of a field, J. Algebra Appl. 11(5) (2012) 1250041, 18 pp.]. The aim of this paper is to deal with the case where [Formula: see text] is any integral domain which is not necessarily a field. Several examples are provided to illustrate our theory.

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