scholarly journals Characterization of multiplication commutative rings with finitely many minimal prime ideals

2019 ◽  
Vol 47 (11) ◽  
pp. 4533-4540 ◽  
Author(s):  
T. Alsuraiheed ◽  
V. V. Bavula
Keyword(s):  
Commutative Rings ◽  
Prime Ideals ◽  
Minimal Prime

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].

Graded 1-absorbing prime ideals of graded commutative rings

2021 ◽  
Author(s):  
Mohammed Issoual
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Graded Ring

Let [Formula: see text] be a group with identity [Formula: see text] and [Formula: see text] be [Formula: see text]-graded commutative ring with [Formula: see text] In this paper, we introduce and study the graded versions of 1-absorbing prime ideal. We give some properties and characterizations of these ideals in graded ring, and we give a characterization of graded 1-absorbing ideal the idealization [Formula: see text]

On Maximal Torsion Radicals

1973 ◽  
Vol 25 (4) ◽  
pp. 712-726 ◽  
Author(s):  
John A. Beachy
Keyword(s):  
General Result ◽  
Noetherian Ring ◽  
Associative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Maximal Torsion ◽  
Minimal Prime

Let R be an associative ring with identity, and let denote the category of unital left R-modules. It is known that if R is a commutative, Noetherian ring, then the maximal torsion radicals of correspond to the minimal prime ideals of R. In fact, Nӑstӑsescu and Popescu [15, Proposition 2.7] have given a more general result valid for arbitrary commutative rings. This paper investigates maximal torsion radicals over rings not necessarily commutative.

scholarly journals Pseudocomplements in groupoids

1978 ◽  
Vol 26 (2) ◽  
pp. 209-219 ◽  
Author(s):  
K. Nirmala Kumari Amma
Keyword(s):  
Boolean Algebra ◽  
Prime Ideals ◽  
Minimal Prime

AbstractThis paper is devoted to a study of pseudocomplements in groupoids. A characterization of an intraregular groupoid is obtained in terms of prime ideals. It is proved that the set of dense elements of an intraregular groupoid S with 0 is the intersection of all the maximal filters of S and that the set of normal elements of an intraregular groupoid closed for pseudocomplements forms a Boolean algebra under natural operations. It is shown that the pseudocomplement of an ideal of an intraregular groupoid with 0 is the intersection of all the minimal prime ideas not containing it.

scholarly journals On Duo Rings

1960 ◽  
Vol 3 (2) ◽  
pp. 167-172 ◽  
Author(s):  
G. Thierrin
Keyword(s):  
Commutative Rings ◽  
Prime Ideals ◽  
Subdirectly Irreducible ◽  
Nilpotent Elements

Following E. H. Feller [l], a ring R is called a duo ring if every one-sided ideal of R is a two-sided ideal.In the first part of this paper, we give some properties of duo rings and we show that the set of the nilpotent elements of a duo ring R is an ideal, the intersection of the completely prime ideals of R.It is easy to see that every duo ring is a subdirect sum of subdirectly irreducible duo rings. We give in the second part of this paper a characterization of the subdirectly irreducible duo rings. This characterization is quite similar to the characterization of the subdirectly irreducible commutative rings, due to N. H. McCoy [2], whose methods we use.

scholarly journals Minimal prime ideals of skew PBW extensions over 2-primal compatible rings

2020 ◽  
Vol 54 (1) ◽  
pp. 39-63
Author(s):  
Mohamed Louzari ◽  
Armando Reyes
Keyword(s):  
Commutative Rings ◽  
Prime Ideals ◽  
Noncommutative Rings ◽  
Polynomial Type ◽  
Ore Extensions ◽  
Skew Pbw Extensions ◽  
Minimal Prime

In this paper, we characterize the units of skew PBW extensions over compatible rings. With this aim, we recall the transfer of the property of being 2-primal for these extensions. As a consequence of our treatment, the results established here generalize those corresponding for commutative rings and Ore extensions of injective type. In this way, we present new results for several noncommutative rings of polynomial type not considered before in the literature.

scholarly journals Spaces of ideals of distributive lattices II. Minimal prime ideals

1974 ◽  
Vol 18 (1) ◽  
pp. 54-72 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Distributive Lattice ◽  
Commutative Rings ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Commutative Semigroups ◽  
Satisfactory State ◽  
Minimal Prime ◽  
The Relationship

This paper, the second of a sequence beginning with [14], deals with the relationship between a distributive lattice L = (L; ∨, ∧, 0) with zero, and certain spaces of minimal prime ideals of L. Similar studies of minimal prime ideals in commutative semigroups [8] and in commutative rings [6] inspired this work, and many of our results are similar to ones in these two articles. However the nature of our situation enables many of these results to be pushed deeper and thus to arrive at a more satisfactory state; indeed with the insight obtained from the simpler lattice situation, one can return to some topics considered in [6], [8] and give complete accounts. We do not do this in the present paper, but leave the details to the reader, see e.g. [15]. Also a study of minimal prime ideals illuminates some topics in the theory of distributive lattices, particularly Stone lattices.

On Maximal Torsion Radicals, II

1975 ◽  
Vol 27 (1) ◽  
pp. 115-120 ◽  
Author(s):  
John A. Beachy
Keyword(s):  
Associative Ring ◽  
Commutative Rings ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Previous Theorem ◽  
Maximal Torsion ◽  
Gabriel Dimension ◽  
Right Noetherian Rings ◽  
Minimal Prime

Let R be an associative ring with identity, and let denote the category of unital left R-modules. The Walkers [6] raised the question of characterizing the maximal torsion radicals of , and showed that if R is commutative and Noetherian, then there is a one-to-one correspondence between maximal torsion radicals and minimal prime ideals of R [6, Theorem 1.29]. Popescu announced [5, Theorem 2.5] that the result remains valid for commutative rings with Gabriel dimension (in the terminology of [2]). Theorem 4.6 below shows that the result holds for rings (not necessarily commutative) with Krull dimension on either the left or right, extending the previous theorem for right Noetherian rings which appeared in [1].

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