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 Some remarks about minimal prime ideals of skew Poincaré-Birkhoff-Witt extensions

2020 ◽  
Vol 30 (2) ◽  
pp. 207-229
Author(s):  
A. Niño ◽  
◽  
A. Reyes ◽  
Keyword(s):  
Prime Ideals ◽  
Noncommutative Rings ◽  
Polynomial Type ◽  
Ore Extensions ◽  
Skew Pbw Extensions ◽  
Minimal Prime

In this paper, we characterize the minimal prime ideals of skew PBW extensions over several classes of rings. We unify different results established in the literature for Ore extensions, and extend all of them to a several families of noncommutative rings of polynomial type which cannot be expressed as these extensions.

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 notion of compatibility for Armendariz and Baer properties over skew PBW extensions

2017 ◽  
pp. 157-178 ◽  
Author(s):  
Armando Reyes ◽  
Héctor Suárez
Keyword(s):  
Universal Enveloping Algebras ◽  
Noncommutative Rings ◽  
Enveloping Algebras ◽  
Ore Extensions ◽  
Skew Pbw Extensions

In this paper we are interested in studying the properties of Armendariz, Baer, quasi-Baer, p.p. and p.q.-Baer over skew PBW extensions. Using a notion of compatibility, we generalize several propositions established for Ore extensions and present new results for several noncommutative rings which can not be expressed as Ore extensions (universal enveloping algebras, diffusion algebras, and others).

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 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.

A NOTE ON COMPLETELY PRIME IDEALS OF ORE EXTENSIONS

2010 ◽  
Vol 20 (03) ◽  
pp. 457-463 ◽  
Author(s):  
V. K. BHAT
Keyword(s):  
Prime Ideal ◽  
Prime Ideals ◽  
Recent Past ◽  
Associated Prime Ideals ◽  
Ore Extensions ◽  
Active Research ◽  
Considerable Work ◽  
Minimal Prime ◽  
Associated Prime ◽  
Completely Prime Ideal

The study of prime ideals has been an area of active research. In recent past a considerable work has been done in this direction. Associated prime ideals and minimal prime ideals of certain types of Ore extensions have been characterized. In this paper a relation between completely prime ideals of a ring R and those of R[x; σ, δ] has been given; σ is an automorphisms of R and δ is a σ-derivation of R. It has been proved that if P is a completely prime ideal of R such that σ(P) = P and δ(P) ⊆ P, then P[x; σ, δ] is a completely prime ideal of R[x; σ, δ]. It has also been proved that this type of relation does not hold for strongly prime ideals.

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