The ℵ1-categoricity of strictly upper triangular matrix rings over algebraically closed fields

1978 ◽  
Vol 43 (2) ◽  
pp. 250-259 ◽  
Author(s):  
Bruce I. Rose
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Closed Field ◽  
Matrix Rings ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Upper Triangular Matrix ◽  
Algebraically Closed Fields

AbstractLet n ≥ 3. The following theorems are proved.Theorem. The theory of the class of strictly upper triangular n × n matrix rings over fields is finitely axiomatizable.Theorem. If R is a strictly upper triangular n × n matrix ring over a field K, then there is a recursive map σ from sentences in the language of rings with constants for K into sentences in the language of rings with constants for R such that K ⊨ φ if and only if R φ σ(φ).Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ1-categorical.

Model theory of strictly upper triangular matrix rings

1980 ◽  
Vol 45 (3) ◽  
pp. 455-463 ◽  
Author(s):  
William H. Wheeler
Keyword(s):  
Model Theory ◽  
Triangular Matrix ◽  
Closed Field ◽  
Matrix Rings ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Closed Fields ◽  
Upper Triangular Matrix ◽  
Algebraically Closed Fields ◽  
Model Completion

Two questions on rings of strictly upper triangular matrices arising from B. Rose's work [5] are answered in this paper. An n × n matrix (αi, j) is strictly upper triangular if αi, j = 0 whenever i ≥ j. The ring of strictly upper triangular n × n matrices with entries from a field F will be denoted by Sn(F). Throughout this paper n will be an integer greater than 2. B. Rose [5] has shown that the complete theory of Sn(F) for an algebraically closed field F is ℵ1categorical. The first main result of this paper is that the rings Sn(F) and Sn(K) for fields F and K are isomorphic or elementarily equivalent if and only if F and K are isomorphic or elementarily equivalent, respectively (Corollary 1.6 and Theorem 2.2). This result shortens the proof of B. Rose's categoricity theorem [5, Theorem 7] by avoiding the co-stability considerations; furthermore, this result yields a proof of the converse of this categoricity theorem. The second main result is that the theory of rings of strictly upper triangular n × n matrices over algebraically closed fields is the model-completion of the theory of rings of strictly upper triangular n × n matrices over arbitrary fields (Theorem 2.5). This answers affirmatively the two conjectures at the end of [5].

Integer-valued polynomials over block matrix algebras

2019 ◽  
Vol 19 (03) ◽  
pp. 2050053
Author(s):  
J. Sedighi Hafshejani ◽  
A. R. Naghipour ◽  
M. R. Rismanchian
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Block Matrix ◽  
Quotient Field ◽  
Matrix Rings ◽  
Matrix Algebras ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Full Matrix Ring ◽  
Upper Triangular Matrix Ring

In this paper, we state a generalization of the ring of integer-valued polynomials over upper triangular matrix rings. The set of integer-valued polynomials over some block matrix rings is studied. In fact, we consider the set of integer-valued polynomials [Formula: see text] for each [Formula: see text], where [Formula: see text] is an integral domain with quotient field [Formula: see text] and [Formula: see text] is a block matrix ring between upper triangular matrix ring [Formula: see text] and full matrix ring [Formula: see text]. In fact, we have [Formula: see text]. It is known that the sets of integer-valued polynomials [Formula: see text] and [Formula: see text] are rings. We state some relations between the rings [Formula: see text] and the partitions of [Formula: see text]. Then, we show that the set [Formula: see text] is a ring for each [Formula: see text]. Further, it is proved that if the ring [Formula: see text] is not Noetherian then the ring [Formula: see text] is not Noetherian, too. Finally, some properties and relations are stated between the rings [Formula: see text], [Formula: see text] and [Formula: see text].

Graded Involutions on Upper-triangular Matrix Algebras

2009 ◽  
Vol 16 (01) ◽  
pp. 103-108 ◽  
Author(s):  
A. Valenti ◽  
M. V. Zaicev
Keyword(s):  
Abelian Group ◽  
Characteristic Zero ◽  
Finite Abelian Group ◽  
Triangular Matrix ◽  
Closed Field ◽  
Triangular Matrices ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Upper Triangular Matrices ◽  
Upper Triangular Matrix

Let UTn be the algebra of n × n upper-triangular matrices over an algebraically closed field of characteristic zero. We describe all G-gradings on UTn by a finite abelian group G commuting with an involution (involution gradings).

scholarly journals Parameter Spaces for Algebraic Equivalence

2017 ◽  
Vol 2019 (6) ◽  
pp. 1863-1893 ◽  
Author(s):  
Jeffrey D Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial
Keyword(s):  
Abelian Varieties ◽  
Closed Field ◽  
Parameter Spaces ◽  
Algebraically Closed Field ◽  
Arbitrary Base ◽  
Closed Fields ◽  
Algebraic Equivalence ◽  
Integral Scheme ◽  
The Difference ◽  
Algebraically Closed Fields

Abstract A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth integral scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families of cycles parameterized by curves, or by abelian varieties. In this article, we extend these results to arbitrary base fields. The strengthening of these results turns out to be a key step in our work elsewhere extending Murre’s results on algebraic representatives for varieties over algebraically closed fields to arbitrary perfect fields.

Projection invariant extending rings

2016 ◽  
Vol 15 (07) ◽  
pp. 1650121 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Adnan Tercan ◽  
Canan C. Yucel
Keyword(s):  
Direct Summand ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Trivial Extension ◽  
Matrix Rings ◽  
Finite Matrix ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
The Right ◽  
Upper Triangular Matrix Ring

A ring [Formula: see text] is said to be right [Formula: see text]-extending if every projection invariant right ideal of [Formula: see text] is essential in a direct summand of [Formula: see text]. In this article, we investigate the transfer of the [Formula: see text]-extending condition between a ring [Formula: see text] and its various ring extensions. More specifically, we characterize the right [Formula: see text]-extending generalized triangular matrix rings; and we show that if [Formula: see text] is [Formula: see text]-extending, then so is [Formula: see text] where [Formula: see text] is an overring of [Formula: see text] which is an essential extension of [Formula: see text], an [Formula: see text] upper triangular matrix ring of [Formula: see text], a column finite or column and row finite matrix ring over [Formula: see text], or a certain type of trivial extension of [Formula: see text].

Presentations of skew fields. I. Existentially closed skew fields and the Nullstellensatz

1975 ◽  
Vol 77 (1) ◽  
pp. 7-19 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Algebraic Geometry ◽  
Matrix Ring ◽  
Commutative Rings ◽  
Commutative Case ◽  
Matrix Rings ◽  
Finite Dimensional ◽  
Existentially Closed ◽  
Closed Fields ◽  
Skew Fields ◽  
Algebraically Closed Fields

1. Introduction. The Nullstellensatz in commutative algebraic geometry may be described as a means of studying certain commutative rings (viz. affine algebras) by their homomorphisms into algebraically closed fields, and a number of attempts have been made to extend the result to the non-commutative case. In particular, Amitsur and Procesi have studied the case of general rings, with homomorphisms into matrix rings over commutative fields ((1), (2)) and Procesi has obtained more precise results for homomorphisms of PI-rings (11). Since a finite-dimensional division algebra can always be embedded in a matrix ring over a field, this includes the case of skew fields that are finite-dimensional over their centre, but it tells us nothing about general skew fields.

scholarly journals Maximal subfields of algebraically closed fields

1980 ◽  
Vol 29 (4) ◽  
pp. 462-468 ◽  
Author(s):  
Robert M. Guralnick ◽  
Michael D. Miller
Keyword(s):  
Galois Group ◽  
Characteristic Zero ◽  
Nonempty Subset ◽  
Rational Numbers ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Algebraically Closed Fields

AbstractLet K be an algebraically closed field of characteristic zero, and S a nonempty subset of K such that S Q = Ø and card S < card K, where Q is the field of rational numbers. By Zorn's Lemma, there exist subfields F of K which are maximal with respect to the property of being disjoint from S. This paper examines such subfields and investigates the Galois group Gal K/F along with the lattice of intermediate subfields.

Remarks on Centers of Rings

2021 ◽  
Vol 28 (01) ◽  
pp. 1-12
Author(s):  
Juan Huang ◽  
Hailan Jin ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Zhelin Piao
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Full Matrix ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

It is proved that for matrices [Formula: see text], [Formula: see text] in the [Formula: see text] by [Formula: see text] upper triangular matrix ring [Formula: see text] over a domain [Formula: see text], if [Formula: see text] is nonzero and central in [Formula: see text] then [Formula: see text]. The [Formula: see text] by [Formula: see text] full matrix rings over right Noetherian domains are also shown to have this property. In this article we treat a ring property that is a generalization of this result, and a ring with such a property is said to be weakly reversible-over-center. The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains. The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally. We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.

scholarly journals Remarks on motivic homotopy theory over algebraically closed fields

2010 ◽  
Vol 7 (1) ◽  
pp. 55-89 ◽  
Author(s):  
Po Hu ◽  
Igor Kriz ◽  
Kyle Ormsby
Keyword(s):  
Homotopy Theory ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Motivic Homotopy Theory ◽  
Stable Homotopy Category ◽  
Motivic Homotopy ◽  
Algebraically Closed Fields

AbstractWe discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

scholarly journals The Variety of Two-dimensional Algebras Over an Algebraically Closed Field

2018 ◽  
Vol 71 (4) ◽  
pp. 819-842 ◽  
Author(s):  
Ivan Kaygorodov ◽  
Yury Volkov
Keyword(s):  
Closed Field ◽  
Two Dimensional ◽  
Algebraically Closed Field ◽  
Irreducible Components ◽  
Closed Fields ◽  
Algebraically Closed Fields

AbstractThe work is devoted to the variety of two-dimensional algebras over algebraically closed fields. First we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of certain algebra series in the variety of two-dimensional algebras. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

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