Extension Of Commutative Rings Research Articles

Rings in which every regular finitely generated ideal is $S$-finitely presented

In this work, we introduce and explore a class of rings where every regular finitely generated ideal is $S$-finitely presented, called a regular $S$-coherent ring. This concept represents a weaker version of the $S$-coherent ring property. It is shown that any $S$-coherent ring is inherently a regular $S$-coherent ring, and in the case of domains, the two properties are equivalent. We also investigate how this notion extends to different settings of commutative ring extensions, including direct products, trivial ring extensions, and the amalgamated duplication of a ring along an ideal. The obtained results yield new examples of regular $S$-coherent rings that are not $S$-coherent.

Noetherian rings of composite generalized power series

Abstract Let A ⊆ B A\subseteq B be an extension of commutative rings with identity, ( S , ≤ ) \left(S,\le ) a nonzero strictly ordered monoid, and S * = S \ { 0 } {S}^{* }\left=S\backslash \left\{0\right\} . Let A + 〚 B S * , ≤ 〛 = { f ∈ 〚 B S , ≤ 〛 ∣ f ( 0 ) ∈ A } A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] =\{f\in [\kern-2pt[ {B}^{S,\le }]\kern-2pt] \hspace{0.33em}| \hspace{0.33em}f\left(0)\in A\} . In this study, we determine when the ring A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring. We prove that when S S is a strict monoid, if A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring, then A A is a Noetherian ring, B B is a finitely generated A A -module, and S S is finitely generated. We also show that if B B is a finitely generated A A -module over a Noetherian ring A A and ( S , ≤ ) \left(S,\le ) is a positive strictly ordered monoid, which is finitely generated, then A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring.

Two absorbing factorization formal power series rings

Let R be a commutative ring with identity. A proper ideal I of R is said to be 2-absorbing if whenever xyz ∈ I for some elements x , y , z ∈ R , then either xy ∈ I or xz ∈ I or yz ∈ I . The ring R is called a two-absorbing factorization ring (TAF-ring) if every proper ideal of R has a two absorbing-factorization that is every ideal is a product of 2-absorbing ideals. In this note, we characterize commutative rings R (respectively, commutative ring extensions A ⊂ B ) for which the ring of formal power series R [ [ X ] ] (respectively, the ring A + XB [ [ X ] ] ) is a TAF-ring.

When every regular ideal is S-finite

In this paper, we introduce a new class of ring called regular S-Noetherian ring, which is a weak version of S-Noetherian ring property. Any S-Noetherian ring is naturally a regular S-Noetherian ring, and in the domain context, these two forms coincide. We study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamated duplication of a ring along an ideal. Our results generate new families of examples of non-S-Noetherian regular S-Noetherian rings.

Commutative Ring Extensions Defined by Perfect-Like Conditions

Commutative Ring Extensions Defined by Perfect-Like Conditions

Distributive FCP extensions

We are dealing with extensions of commutative rings [Formula: see text] whose chains of the poset [Formula: see text] of their subextensions are finite (i.e. [Formula: see text] has the FCP property) and such that [Formula: see text] is a distributive lattice, that we call distributive FCP extensions. Note that the lattice [Formula: see text] of a distributive FCP extension is finite. This paper is the continuation of our earlier papers where we studied catenarian and Boolean extensions. Actually, for an FCP extension, the following implications hold: Boolean [Formula: see text] distributive [Formula: see text] catenarian. A comprehensive characterization of distributive FCP extensions actually remains a challenge, essentially because the same problem for field extensions is not completely solved. Nevertheless, we are able to exhibit a lot of positive results for some classes of extensions. A main result is that an FCP extension [Formula: see text] is distributive if and only if [Formula: see text] is distributive, where [Formula: see text] is the integral closure of [Formula: see text] in [Formula: see text]. A special attention is paid to distributive field extensions.

On a weak version of S-Noetherianity

In this paper, we introduce a new class of ring called [Formula: see text]-[Formula: see text]-Noetherian ring, which is a weak version of [Formula: see text]-Noetherian ring property and study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamation of rings. Furthermore, we define the concept of nonnil [Formula: see text]-[Formula: see text]-Noetherian ring property which is a generalization of the [Formula: see text]-[Formula: see text]-Noetherian domain property and establish a characterization of this notion using pullbacks.

Commutative ring extensions defined by perfect-like conditions

UDC 512.5 In 2005, Enochs, Jenda, and López-Romos extended the notion of perfect rings to n -perfect rings such that a ring is n -perfect if every flat module has projective dimension less or equal than n . Later, Jhilal and Mahdou defined a commutative unital ring R to be strongly n -perfect if any R -module of flat dimension less or equal than n has a projective dimension less or equal than n . Recently Purkait defined a ring R to be n -semiperfect if R ¯ = R / R a d ( R ) is semisimple and n -potents lift modulo R a d ( R ) . We study of three classes of rings, namely, n -perfect, strongly n -perfect, and n -semiperfect rings. We investigate these notions in several ring-theoretic structures with an aim of construction of new original families of examples satisfying the indicated properties and subject to various ring-theoretic properties.

Classes of trivial ring extensions and amalgamations subject to pseudo-almost valuation condition

Abstract This paper studies the transfer of pseudo-almost valuation property (PAVR property for short) to various context of commutative ring extensions such as power series ring, trivial ring extension and amalgamation. Our work is motivated by an attempt to generate new original classes of rings satisfying this property. The obtained results are backed with new and illustrative examples arising as trivial ring extensions, amalgamations and pullback constructions.

Certain towers of ramified minimal ring extensions of commutative rings, II

Let [Formula: see text] be an odd prime number and [Formula: see text]. Assume that either [Formula: see text]i[Formula: see text] [Formula: see text] or [Formula: see text]ii[Formula: see text] [Formula: see text] and [Formula: see text] is congruent to either [Formula: see text] or [Formula: see text] modulo [Formula: see text] [Formula: see text]respectively, assume that [Formula: see text] and [Formula: see text] is congruent to either [Formula: see text] or [Formula: see text] modulo [Formula: see text]. Then there exist exactly five [Formula: see text]respectively, exactly seven[Formula: see text] isomorphism classes of rings [Formula: see text] for which there exists a tower [Formula: see text] of ramified [Formula: see text]integral minimal[Formula: see text] ring extensions such that [Formula: see text] is the only ring properly contained between [Formula: see text] and [Formula: see text]. We produce a set of isomorphism class representatives of such [Formula: see text] and, for each such [Formula: see text], we show that, up to isomorphism, the corresponding ring [Formula: see text] is the idealization [Formula: see text]. One consequence, for each integer [Formula: see text] whose prime-power factorization [Formula: see text] (with pairwise distinct prime numbers [Formula: see text] and positive integers [Formula: see text]) satisfies [Formula: see text] for all [Formula: see text], is a classification, up to isomorphism, of the rings that have cardinality [Formula: see text] and exactly two proper (unital) subrings.

WHEN PRIME SUBMODULES ARE PRIME IDEALS?

"We study commutative ring extensions R ⊂ S in which every ideal of S that is a prime R-submodule of S is prime."

On the Commutative Ring Extensions with at Most Two Non Prüfer Intermediate Rings

On the Commutative Ring Extensions with at Most Two Non Prüfer Intermediate Rings

THE LOEWY SERIES OF AN FCP (DISTRIBUTIVE) RING EXTENSION

If $R\subseteq S$ is an extension of commutative rings, we consider the lattice $([R,S],\subseteq)$ of all the $R$-subalgebras of $S$.
 We assume that the poset $[R,S]$ is both Artinian and Noetherian; that is, $R\subseteq S$ is an FCP extension.
 The Loewy series of such lattices are studied. Most of main results are gotten in case these posets are distributive,
 which occurs for integrally closed extensions. In general, the situation is much more complicated. We give a discussion for finite field extensions.

Partial generalized crossed products and a seven-term exact sequence

For a partial Galois extension of commutative rings we give a seven term exact sequence which generalize the Chase-Harrison-Rosenberg sequence.

Partial actions and cyclic kummer’s theory

We introduce a theory of cyclic Kummer extensions of commutative rings for partial Galois extensions of finite groups, extending some of the well-known results of the theory of Kummer extensions of commutative rings developed by Borevich. In particular, we provide necessary and sufficient conditions to determine when a partial [Formula: see text]-Kummerian extension is equivalent to either a radical or an [Formula: see text]-radical extension, for some subgroup [Formula: see text] of the cyclic group [Formula: see text].

Ring Extensions with Finitely Many Non-Artinian Intermediate Rings

The commutative ring extensions with exactly two non-Artinian intermediate rings are characterized. An initial step involves the description of the commutative ring extensions with only one non-Artinian intermediate ring.

Some commutative ring extensions defined by almost Bézout condition

In this paper, we study the almost Bézout property in different commutative ring extensions, namely, in bi-amalgamated algebras and pairs of rings. In Section 2, we deal with almost Bézout domains issued from bi-amalgamations. Our results capitalize well known results on amalgamations and pullbacks as well as generate new original class of rings satisfying this property. Section 3 investigates pairs of rings where all intermediate rings are almost Bézout domains. As an application of our results, we characterize pairs of rings $(R,T)$, where $R$ arises from a $(T,M,D)$ construction to be an almost Bézout domain.

Radical factorization for trivial extensions and amalgamated duplication rings

Let [Formula: see text] be a commutative ring extension such that [Formula: see text] is a trivial extension of [Formula: see text] (denoted by [Formula: see text]) or an amalgamated duplication of [Formula: see text] along some ideal of [Formula: see text] (denoted by [Formula: see text]. This paper examines the transfer of AM-ring, N-ring, SSP-ring and SP-ring between [Formula: see text] and [Formula: see text]. We study the transfer of those properties to trivial ring extension. Call a special SSP-ring an SSP-ring of the following type: it is the trivial extension of [Formula: see text] by a C-module [Formula: see text], where [Formula: see text] is an SSP-ring, [Formula: see text] a von Neumann regular ring and [Formula: see text] a multiplication C-module. We show that every SSP-ring with finitely many minimal primes which is a trivial extension is in fact special. Furthermore, we study the transfer of the above properties to amalgamated duplication along an ideal with some extra hypothesis. Our results allows us to construct nontrivial and original examples of rings satisfying the above properties.

Boolean FIP ring extensions

We characterize extensions of commutative rings whose sets of subextensions [R, S] are finite (i.e. has the FIP property) and are Boolean lattices, that we call Boolean FIP extensions. Some characterizations involve “factorial” properties of the poset [R, S]. A non-trivial result is that each subextension of a Boolean FIP extension is simple (i.e. is a simple pair).

Composite Hurwitz Rings as PF-Rings and PP-Rings

Let R ⊆ T be an extension of commutative rings with identity and H ( R , T ) (respectively, h ( R , T ) ) the composite Hurwitz series ring (respectively, composite Hurwitz polynomial ring). In this article, we study equivalent conditions for the rings H ( R , T ) and h ( R , T ) to be PF-rings and PP-rings. We also give some examples of PP-rings and PF-rings via the rings H ( R , T ) and h ( R , T ) .