scholarly journals Galois Closures of Non-commutative Rings and an Application to Hermitian Representations

2018 ◽  
Vol 2020 (21) ◽  
pp. 7944-7974
Author(s):  
Wei Ho ◽  
Matthew Satriano
Keyword(s):  
Invariant Theory ◽  
Commutative Rings ◽  
Base Change ◽  
Product Formula ◽  
Ring Extensions ◽  
Galois Closures

Abstract Galois closures of commutative rank $n$ ring extensions were introduced by Bhargava and the 2nd author. In this paper, we generalize the construction to the case of non-commutative rings. We show that noncommutative Galois closures commute with base change and satisfy a product formula. As an application, we give a uniform construction of many of the representations arising in arithmetic invariant theory, including many Vinberg representations.

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.

Almost Bézout rings and almost GCD-rings

2019 ◽  
Vol 13 (06) ◽  
pp. 2050107
Author(s):  
Abdelhaq El Khalfi ◽  
Najib Mahdou
Keyword(s):  
Commutative Rings ◽  
Ring Extensions

In this paper, we study the possible transfer of the property of being an [Formula: see text]-ring to trivial ring extensions and amalgamated algebras along an ideal. Also, we extend the notion of an almost GCD-domain to the context of arbitrary rings, and we study the possible transfer of this notion to trivial ring extensions and amalgamated algebras along an ideal. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

scholarly journals Operads and Γ-homology of commutative rings

2002 ◽  
Vol 132 (2) ◽  
pp. 197-234 ◽  
Author(s):  
ALAN ROBINSON ◽  
SARAH WHITEHOUSE
Keyword(s):  
Homotopy Theory ◽  
Commutative Rings ◽  
Homology Theory ◽  
Base Change ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
General Properties ◽  
Quillen Homology

We introduce Γ-homology, the natural homology theory for E∞-algebras, and a cyclic version of it. Γ-homology specializes to a new homology theory for discrete commutative rings, very different in general from André–Quillen homology. We prove its general properties, including at base change and transitivity theorems. We give an explicit bicomplex for the Γ-homology of a discrete algebra, and elucidate connections with stable homotopy theory.

Rings in which every regular locally principal ideal is projective

2019 ◽  
Vol 56 (2) ◽  
pp. 241-251
Author(s):  
Rachida El Khalfaoui ◽  
Najib Mahdou
Keyword(s):  
Principal Ideal ◽  
Commutative Rings ◽  
Direct Products ◽  
Ring Extensions

Abstract In this article, we study the class of rings in which every regular locally principal ideal is projective called LPP-rings. We investigate the transfer of this property to various constructions such as direct products, amalgamation of rings, and trivial ring extensions. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned property.

On PM-rings, rings of finite character and h-local rings

2014 ◽  
Vol 13 (06) ◽  
pp. 1450018
Author(s):  
N. Mahdou ◽  
A. Mimouni ◽  
M. A. S. Moutui
Keyword(s):  
Commutative Rings ◽  
Local Rings ◽  
Local Properties ◽  
Ring Extensions

In this paper, we investigate the transfer of the notions of pm-rings, rings of finite character and h-local rings to trivial ring extensions of rings by modules, amalgamations of rings along ideals and pullbacks. Our aim is to provide new classes of commutative rings satisfying these properties and our results generate new families of examples of rings for which the finite character and semi-local properties are equivalents.

Ring extensions of length two

2019 ◽  
Vol 18 (09) ◽  
pp. 1950174 ◽  
Author(s):  
Gabriel Picavet ◽  
Martine Picavet-L’Hermitte
Keyword(s):  
Commutative Algebra ◽  
Commutative Rings ◽  
Field Extension ◽  
Ring Extensions

We characterize extensions of commutative rings [Formula: see text] such that [Formula: see text] is minimal for each [Formula: see text]-subalgebra [Formula: see text] of [Formula: see text] with [Formula: see text]. This property is equivalent to [Formula: see text] has length 2. Such extensions are either pointwise minimal or simple. We are able to compute the number of subextensions of [Formula: see text]. Besides commutative algebra considerations, our main result uses the concept of principal subfields of a finite separable field extension, which was recently introduced by van Hoeij et al. As a corollary of this paper, we get that simple extensions of length 2 have FIP.

scholarly journals Commutativity of the Haagerup tensor product and base change for operator modules

2021 ◽  
pp. 1-11
Author(s):  
Tyrone Crisp
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Base Change ◽  
Product Formula ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Continuous Mappings ◽  
Continuous Fields ◽  
Bounded Norm

By computing the completely bounded norm of the flip map on the Haagerup tensor product [Formula: see text] associated to a pair of continuous mappings of locally compact Hausdorff spaces [Formula: see text], we establish a simple characterization of the Beck-Chevalley condition for base change of operator modules over commutative [Formula: see text]-algebras, and a descent theorem for continuous fields of Hilbert spaces.

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