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.

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 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.

Commutative rings and modules that are Nil*-coherent or special Nil*-coherent

2017 ◽  
Vol 16 (10) ◽  
pp. 1750187 ◽  
Author(s):  
Karima Alaoui Ismaili ◽  
David E. Dobbs ◽  
Najib Mahdou
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Basic Properties ◽  
Unital Ring ◽  
Reduced Ring ◽  
Coherent Ring ◽  
Ring Extensions ◽  
Coherent Rings ◽  
Finitely Presented

Recently, Xiang and Ouyang defined a (commutative unital) ring [Formula: see text] to be Nil[Formula: see text]-coherent if each finitely generated ideal of [Formula: see text] that is contained in Nil[Formula: see text] is a finitely presented [Formula: see text]-module. We define and study Nil[Formula: see text]-coherent modules and special Nil[Formula: see text]-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil[Formula: see text]-coherent ring and any special Nil[Formula: see text]-coherent ring is a Nil[Formula: see text]-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil[Formula: see text]-coherent ring (regardless of whether it is coherent). Several examples of Nil[Formula: see text]-coherent rings that are not special Nil[Formula: see text]-coherent rings are obtained as byproducts of our study of the transfer of the Nil[Formula: see text]-coherent and the special Nil[Formula: see text]-coherent properties to trivial ring extensions and amalgamated algebras.

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.

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.

Strong Separativity over Regular Rings

2015 ◽  
Vol 22 (03) ◽  
pp. 413-420
Author(s):  
Huanyin Chen
Keyword(s):  
Finitely Generated ◽  
Regular Ideal ◽  
Regular Rings

An ideal I of a ring R is strongly separative provided that for all finitely generated projective R-modules A, B with A=AI and B=BI, if 2A ≅ A ⊕ B, then A ≅ B. We prove in this paper that a regular ideal I of a ring R is strongly separative if and only if each a ∈ 1+I satisfying (1-a)R ∝ r(a) is unit-regular, if and only if each a ∈ 1+I satisfying (1-a2)R ∝ r(a2) is unit-regular, if and only if each a ∈ 1+I satisfying R(1-a)R=R r(a) is unit-regular, if and only if each a ∈ 1+I satisfying R(1-a2)R=R r(a2) is unit-regular.

scholarly journals Galois Theory for Rings with Finitely Many Idempotents

1966 ◽  
Vol 27 (2) ◽  
pp. 721-731 ◽  
Author(s):  
O. E. Villamayor ◽  
D. Zelinsky
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Fundamental Theorem ◽  
Galois Theory ◽  
Finitely Generated ◽  
Fixed Ring ◽  
Ring Extensions ◽  
A Finite Group

In [5], Chase, Harrison and Rosenberg proved the Fundamental Theorem of Galois Theory for commutative ring extensions S ⊃ R under two hypotheses: (i) 5 (and hence R) has no idempotents except 0 and l; and (ii) 5 is Galois over R with respect to a finite group G—which in the presence of (i) is equivalent to (ii′): S is separable as an R-algebra, finitely generated and projective as an R-module, and the fixed ring under the group of all R-algebra automorphisms of S is exactly R.

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.

Two notes on ideal-transforms

1987 ◽  
Vol 102 (3) ◽  
pp. 389-397 ◽  
Author(s):  
Daniel Katz ◽  
Louis J. Ratliff
Keyword(s):  
Large Class ◽  
Regular Element ◽  
Finitely Generated ◽  
Closed Set ◽  
Regular Elements ◽  
Integral Extension ◽  
Regular Ideal ◽  
Integrally Closed ◽  
The Ideal

AbstractThe first note gives two new characterization of the ideal-transform T(I) of a finitely generated regular ideal I in a large class of rings. Specifically, if b is a regular element in I, then there exists a regular element c ∈ I and a multiplicatively closed set S of regular elements in R such that T(I) = T((b, c)R) = Rb ∩ Rc = Rb ∩ Rs, so T(I) is the ideal-transform of an ideal generated by two elements, and every ring of the form Rb ∩ Rs is an ideal-transform. The second theorem shows that if T(I) is integrally closed, then it is a Krull ring. As an application of these results we strengthen some known results concerning when certain ideal-transforms of the Rees ring R(R, I) are finite or integral extension rings of R(R, I).

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