Multiplicative canonical ideals of ring extensions

2017 ◽  
Vol 16 (04) ◽  
pp. 1750069 ◽  
Author(s):  
Simplice Tchamna
Keyword(s):  
Ring Extension ◽  
Ideal Formula ◽  
Regular Ideal ◽  
Canonical Ideal ◽  
Ring Extensions

We study properties of multiplicative canonical (m-canonical) ideals of ring extensions. Let [Formula: see text] be a ring extension. A nonzero [Formula: see text]-regular ideal [Formula: see text] of [Formula: see text] is called an m-canonical ideal of the extension [Formula: see text] if [Formula: see text] for all [Formula: see text]-regular ideal [Formula: see text] of [Formula: see text]. We study m-canonical ideals for pullback diagrams, and we use the notion of m-canonical ideal to characterize Prüfer extensions.

scholarly journals Cancellation ideals of a ring extension

2021 ◽  
Vol 32 (1) ◽  
pp. 138-146
Author(s):  
S. Tchamna ◽  
Keyword(s):  
Ring Extension ◽  
Finitely Generated ◽  
Regular Ideal ◽  
Ring Extensions

We study properties of cancellation ideals of ring extensions. Let R⊆S be a ring extension. A nonzero S-regular ideal I of R is called a (quasi)-cancellation ideal of the ring extension R⊆S if whenever IB=IC for two S-regular (finitely generated) R-submodules B and C of S, then B=C. We show that a finitely generated ideal I is a cancellation ideal of the ring extension R⊆S if and only if I is S-invertible.

Local dimension of trivial extension and amalgamation of rings

2021 ◽  
Author(s):  
Rachida El Khalfaoui ◽  
Najib Mahdou ◽  
Siamak Yassemi
Keyword(s):  
Homomorphic Image ◽  
Trivial Extension ◽  
Local Dimension ◽  
Ring Extension ◽  
Ring Extensions

Local dimension is an ordinal valued invariant that is in some sense a measure of how far a ring is from being local and denoted [Formula: see text]. The purpose of this paper is to study the local dimension of ring extensions such as homomorphic image, trivial ring extension and the amalgamation of rings.

A note on the ascending chain condition of ideals

2019 ◽  
Vol 19 (07) ◽  
pp. 2050135 ◽  
Author(s):  
Ibrahim Al-Ayyoub ◽  
Malik Jaradat ◽  
Khaldoun Al-Zoubi
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Chain Condition ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Regular Ideal ◽  
Reduction Number ◽  
Number Formula ◽  
A Chain ◽  
Ascending Chain Condition

We construct ascending chains of ideals in a commutative Noetherian ring [Formula: see text] that reach arbitrary long sequences of equalities, however the chain does not become stationary at that point. For a regular ideal [Formula: see text] in [Formula: see text], the Ratliff–Rush reduction number [Formula: see text] of [Formula: see text] is the smallest positive integer [Formula: see text] at which the chain [Formula: see text] becomes stationary. We construct ideals [Formula: see text] so that such a chain reaches an arbitrary long sequence of equalities but [Formula: see text] is not being reached yet.

Pure ring extensions and self FP-injective rings

1989 ◽  
Vol 105 (3) ◽  
pp. 447-458 ◽  
Author(s):  
P. Menal ◽  
P. Vámos
Keyword(s):  
Additive Group ◽  
Ring Extension ◽  
Counter Example ◽  
Existentially Closed ◽  
Ring Extensions ◽  
Algebra Over A Field

In this paper we investigate the following problem: is a ring R right self FP-injective if it has the property that it is pure, as a right R-module, in every ring extension? The answer is ‘almost always’; for example it is ‘yes’ when R is an algebra over a field or its additive group has no torsion. A counter-example is provided to show that the answer is ‘no’ in general. Rings which are pure in all ring extensions were studied by Sabbagh in [4] where it is pointed out that existentially closed rings have this property. Therefore every ring embeds in such a ring. We will show that the weaker notion of being weakly linearly existentially closed is equivalent to self FP-injectivity. As a consequence we obtain that any ring embeds in a self FP-injective ring.

scholarly journals On free ring extensions of degreen

1981 ◽  
Vol 4 (4) ◽  
pp. 703-709
Author(s):  
George Szeto
Keyword(s):  
Galois Extension ◽  
Quaternion Algebra ◽  
Ring Extension ◽  
Quaternion Algebras ◽  
Free Ring ◽  
Azumaya Algebra ◽  
Ring Extensions ◽  
One To One ◽  
Generalized Quaternion

Nagahara and Kishimoto [1] studied free ring extensionsB(x)of degreenfor some integernover a ringBwith 1, wherexn=b,cx=xρ(c)for allcand somebinB(ρ=automophism of  B), and{1,x…,xn−1}is a basis. Parimala and Sridharan [2], and the author investigated a class of free ring extensions called generalized quaternion algebras in whichb=−1andρis of order 2. The purpose of the present paper is to generalize a characterization of a generalized quaternion algebra to a free ring extension of degreenin terms of the Azumaya algebra. Also, it is shown that a one-to-one correspondence between the set of invariant ideals ofBunderρand the set of ideals ofB(x)leads to a relation of the Galois extensionBover an invariant subring underρto the center ofB.

scholarly journals On ring extensions of FSG rings

1994 ◽  
Vol 49 (3) ◽  
pp. 365-371 ◽  
Author(s):  
Le Van Thuyet
Keyword(s):  
Group Rings ◽  
Ring Extension ◽  
Finitely Generated ◽  
Ring Extensions

A ring R is called right FSG if every finitely generated right R-subgenerator is a generator. In this note we consider the question of when a ring extension of a given right FSG ring is right FSG and the converse. As a consequence we obtain some results about right FSG group rings.

Trivial extensions defined by 2-absorbing-like conditions

2018 ◽  
Vol 17 (11) ◽  
pp. 1850208 ◽  
Author(s):  
Mohammed Issoual ◽  
Najib Mahdou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Ideal Formula ◽  
Ring Extensions

Let [Formula: see text] be a commutative ring with [Formula: see text] The notions of 2-absorbing ideal and 2-absorbing primary ideal are introduced by Ayman Badawi as generalizations of prime ideal and primary ideal, respectively. A proper ideal [Formula: see text] of [Formula: see text] is called a 2-absorbing ideal of [Formula: see text] (respectively, 2-absorbing primary ideal) if whenever [Formula: see text] with [Formula: see text] then [Formula: see text] or [Formula: see text] or [Formula: see text] (respectively, [Formula: see text] or [Formula: see text] or [Formula: see text]). In this paper, we investigate the transfer of 2-absorbing-like properties to trivial ring extensions.

On n-absorbing ideals and (m,n)-closed ideals in trivial ring extensions of commutative rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950123 ◽  
Author(s):  
Ayman Badawi ◽  
Mohammed Issoual ◽  
Najib Mahdou
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
General Concept ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ring Extension ◽  
Text Recall ◽  
Ideal Formula

Let [Formula: see text] be a commutative ring with [Formula: see text]. Recall that a proper ideal [Formula: see text] of [Formula: see text] is called a 2-absorbing ideal of [Formula: see text] if [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] or [Formula: see text]. A more general concept than 2-absorbing ideals is the concept of [Formula: see text]-absorbing ideals. Let [Formula: see text] be a positive integer. A proper ideal [Formula: see text] of [Formula: see text] is called an n-absorbing ideal of [Formula: see text] if [Formula: see text] and [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose product is in [Formula: see text]. The concept of [Formula: see text]-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of [Formula: see text] is a 1-absorbing ideal of [Formula: see text]). Let [Formula: see text] and [Formula: see text] be integers with [Formula: see text]. A proper ideal [Formula: see text] of [Formula: see text] is called an [Formula: see text]-closed ideal of [Formula: see text] if whenever [Formula: see text] for some [Formula: see text] implies [Formula: see text]. Let [Formula: see text] be a commutative ring with [Formula: see text] and [Formula: see text] be an [Formula: see text]-module. In this paper, we study [Formula: see text]-absorbing ideals and [Formula: see text]-closed ideals in the trivial ring extension of [Formula: see text] by [Formula: see text] (or idealization of [Formula: see text] over [Formula: see text]) that is denoted by [Formula: see text].

scholarly journals Normal Bases on Galois Ring Extensions

2018 ◽  
Author(s):  
Zhang Aixian ◽  
Feng Keqin
Keyword(s):  
Finite Field ◽  
Field Extension ◽  
Galois Ring ◽  
Normal Basis ◽  
Ring Extension ◽  
Normal Bases ◽  
Ring Extensions

In this paper we study the normal bases for Galois ring extension ${{R}} / {Z}_{p^r}$ where ${R}$ = ${GR}$(pr, n). We present a criterion on normal basis for ${{R}} / {Z}_{p^r}$ and reduce this problem to one of finite field extension $\overline{R} / \overline{Z}_{p^r}=F_{q} / F_{p}  (q=p^n)$ by Theorem 1. We determine all optimal normal bases for Galois ring extension.

MORITA DUALITY AND RING EXTENSIONS

2012 ◽  
Vol 12 (02) ◽  
pp. 1250160
Author(s):  
KAZUTOSHI KOIKE
Keyword(s):  
Finite Ring ◽  
Ring Extension ◽  
Semigroup Rings ◽  
Self Duality ◽  
Category Equivalence ◽  
Ring Extensions

In this paper, we show that there exists a category equivalence between certain categories of A-rings (respectively, ring extensions of A) and B-rings (respectively, ring extensions of B), where A and B are Morita dual rings. In this category equivalence, corresponding two A-ring and B-ring are Morita dual. This is an improvement of a result of Müller, which state that if a ring A has a Morita duality induced by a bimodule BQA and R is a ring extension of A such that RA and Hom A(R, Q)A are linearly compact, then R has a Morita duality induced by the bimodule S End R( Hom A(R, Q))R with S = End R( Hom A(R, Q)). We also investigate relationships between Morita duality and finite ring extensions. Particularly, we show that if A and B are Morita dual rings with B basic, then every finite triangular (respectively, normalizing) extension R of A is Morita dual to a finite triangular (respectively, normalizing) extension S of B, and we give a result about finite centralizing free extensions, which unify a result of Mano about self-duality and a result of Fuller–Haack about semigroup rings.

Sign in / Sign up

Export Citation Format

Share Document