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.

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.

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.

Radical factorization for trivial extensions and amalgamated duplication rings

2020 ◽  
pp. 2150025
Author(s):  
Tiberiu Dumitrescu ◽  
Najib Mahdou ◽  
Youssef Zahir
Keyword(s):  
Commutative Ring ◽  
Regular Ring ◽  
Trivial Extension ◽  
Ring Extension ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Amalgamated Duplication

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.

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.

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.

scholarly journals On the construction of ring extensions

1976 ◽  
Vol 17 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Edward L. Green ◽  
Idun Reiten
Keyword(s):  
Trivial Extension ◽  
Artin Ring ◽  
Ring Extensions

Let ∧ denote a basic artin ring and r its radical. In most of this paper we assume that r2 = 0 and that Λ is a trivial extension Λ/r ⋉ r (see Section 1 for definition). Let P1 …, Pn be the non-isomorphic indecomposable projective (left) Λ-modules, and consider triples (Pi, Mi, ui), where the Mi, are (left) Λ-modules and ui:rPi → Mi/rMi isomorphisms. From this data we construct a new ring Г, which in “nice cases” has the property that r′3 = 0, Г/r′2 ≅ Λ, and r′ Qi ≅ Mi as (left) Λ-modules, where the Qi are the indecomposable projective (left) Г-modules and r′ is the radical of Г.

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.

Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

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