Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial

2006 ◽  
Vol 49 (2) ◽  
pp. 265-269 ◽  
Author(s):  
W. K. Nicholson ◽  
Y. Zhou
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Arbitrary Ring ◽  
Semisimple Module ◽  
Countable Dimension

AbstractIf C = C(R) denotes the center of a ring R and g(x) is a polynomial in C[x], Camillo and Simón called a ring g(x)-clean if every element is the sum of a unit and a root of g(x). If V is a vector space of countable dimension over a division ring D, they showed that end DV is g(x)-clean provided that g(x) has two roots in C(D). If g(x) = x – x2 this shows that end DV is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that end RM is g(x)-clean for any semisimple module M over an arbitrary ring R provided that g(x) ∈ (x – a)(x – b)C[x] where a, b ∈ C and both b and b – a are units in R.

scholarly journals Rings in which every element is either a sum or a difference of a nilpotent and an idempotent

2016 ◽  
Vol 15 (08) ◽  
pp. 1650148 ◽  
Author(s):  
Simion Breaz ◽  
Peter Danchev ◽  
Yiqiang Zhou
Keyword(s):  
Division Ring ◽  
Matrix Ring ◽  
Decomposition Theorem ◽  
Arbitrary Ring ◽  
Clean Rings ◽  
Clean Ring ◽  
Unital Rings ◽  
Nil Clean Ring ◽  
Comprehensive Study

Generalizing the notion of nil-cleanness from [A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211], in parallel to [P. V. Danchev and W. Wm. McGovern, Commutative weakly nil clean unital rings, J. Algebra 425 (2015) 410–422], we define the concept of weak nil-cleanness for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is weakly nil-clean. It is also completely determined when a matrix ring over a division ring is weakly nil-clean.

FINITARY AUTOMORPHISM GROUPS OVER NON-COMMUTATIVE RINGS

2001 ◽  
Vol 64 (3) ◽  
pp. 611-623 ◽  
Author(s):  
B. A. F. WEHRFRITZ
Keyword(s):  
Division Ring ◽  
Automorphism Groups ◽  
Commutative Rings ◽  
Linear Groups ◽  
Locally Finite ◽  
Arbitrary Ring ◽  
Finitary Automorphism

The notion of a group of finitary automorphisms of an arbitrary module over an arbitrary ring is introduced, and it is shown how properties of such groups can be derived from the case where the ring is a division ring (that is, from the properties of finitary skew linear groups). The results are particularly strong if either the group is locally finite or the module is Noetherian.

VECTOR SPACE GENERATED BY THE MULTIPLICATIVE COMMUTATORS OF A DIVISION RING

2013 ◽  
Vol 12 (08) ◽  
pp. 1350043 ◽  
Author(s):  
M. AGHABALI ◽  
S. AKBARI ◽  
M. ARIANNEJAD ◽  
A. MADADI
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Characteristic Zero

Let D be a division ring with center F. An element of the form xyx-1y-1 ∈ D is called a multiplicative commutator. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In this paper it is shown that if D is algebraic over F and Char (D) = 0, then D = T(D). We conjecture that it is true in general. Among other results it is shown that in characteristic zero if T(D) is algebraic over F, then D is algebraic over F.

scholarly journals On subgroups in division rings of type 2

2012 ◽  
Vol 49 (4) ◽  
pp. 549-557
Author(s):  
Bui Hai ◽  
Trinh Deo ◽  
Mai Bien
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Division Rings ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Multiplicative Subgroups

Let D be a division ring with center F. We say that D is a division ring of type 2 if for every two elements x, y ∈ D, the division subring F(x, y) is a finite dimensional vector space over F. In this paper we investigate multiplicative subgroups in such a ring.

scholarly journals A note on multiplicative commutators of division rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950031
Author(s):  
Roozbeh Hazrat
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Commutator Subgroup ◽  
Division Rings

We give an example of a division ring [Formula: see text] whose multiplicative commutator subgroup does not generate [Formula: see text] as a vector space over its center, thus disproving the conjecture posed in [M. Aghabali, S. Akbari, M. Ariannejad and A. Madadi, Vector space generated by the multiplicative commutators of a division ring, J. Algebra Appl. 12(8) (2013) 7 pp.].

scholarly journals LINEAR DIVISION RINGS

2009 ◽  
Vol 12 (17) ◽  
pp. 5-11
Author(s):  
Bien Hoang Mai ◽  
Hai Xuan Bui
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Finite Subset ◽  
Multiplicative Group ◽  
Dimensional Vector ◽  
Finitely Generated ◽  
Dimensional Vector Space ◽  
Division Rings ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Let D be a division ring with the center F and suppose that D* is the multiplicative group of D. D is called centrally finite if D is a finite dimensional vector space over F and D is locally centrally finite if every finite subset of D generates over F a division subring which is a finite dimensional vector space over F. We say that D is a linear division ring if every finite subset of D generates over Fa centrally finite division subring. It is obvious that every locally centrally finite division ring is linear. In this report we show that the inverse is not true by giving an example of a linear division ring which is not locally centrally finite. Further, we give some properties of subgroups in linear division rings. In particular, we show that every finitely generated subnormal subgroup in a linear ring is central. An interesting corollary is obtained as the following: If D is a linear division ring and D* is finitely generated, then D is a finite field.

scholarly journals Countable vector lattices

1974 ◽  
Vol 10 (3) ◽  
pp. 371-376 ◽  
Author(s):  
Paul F. Conrad
Keyword(s):  
Vector Space ◽  
Vector Lattice ◽  
Real Vector Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Real Numbers ◽  
Vector Lattices ◽  
Direct Sum ◽  
Ordered Vector Spaces ◽  
Countable Dimension

In his paper “On the structure of ordered real vector spaces” (Publ. Math. Debrecen 4 (1955–56), 334–343), Erdös shows that a totally ordered real vector space of countable dimension is order isomorphic to a lexicographic direct sum of copies of the group of real numbers. Brown, in “Valued vector spaces of countable dimension” (Publ. Math. Debrecen 18 (1971), 149–151), extends the result to a valued vector space of countable dimension and greatly simplifies the proof. In this note it is shown that a finite valued vector lattice of countable dimension is order isomorphic to a direct sum of o–simple totally ordered vector spaces. One obtains as corollaries the result of Erdös and the applications that Brown makes to totally ordered spaces.

Strongly and co-strongly minimal abelian structures

2010 ◽  
Vol 75 (2) ◽  
pp. 442-458 ◽  
Author(s):  
Ehud Hrushovski ◽  
James Loveys
Keyword(s):  
Vector Space ◽  
Finite Index ◽  
Division Ring ◽  
First Case ◽  
Special Cases ◽  
Structure Theorems ◽  
Algebraic Element ◽  
Minimal Module ◽  
Induced Structure

AbstractWe give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);2. when the theory of the structure is strongly minimal.In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d ∈ D. the index of A ∩ dA, in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.

Density of Polynomial Maps

2010 ◽  
Vol 53 (2) ◽  
pp. 223-229 ◽  
Author(s):  
Chen-Lian Chuang ◽  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Vector Space ◽  
Prime Ring ◽  
Division Ring ◽  
Polynomial Identity ◽  
Matrix Ring ◽  
Polynomial Maps ◽  
Finite Matrix ◽  
Nonzero Polynomial

AbstractLet R be a dense subring of End(DV), where V is a left vector space over a division ring D. If dimDV = ∞, then the range of any nonzero polynomial ƒ (X1, … , Xm) on R is dense in End(DV). As an application, let R be a prime ring without nonzero nil one-sided ideals and 0 ≠ a ∈ R. If a f (x1, … , xm)n(xi) = 0 for all x1, … , xm ∈ R, where n(xi ) is a positive integer depending on x1, … , xm, then ƒ (X1, … , Xm) is a polynomial identity of R unless R is a finite matrix ring over a finite field.

Decompositions of Linear Transformations over Division Rings

2012 ◽  
Vol 19 (03) ◽  
pp. 459-464 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Linear Transformations ◽  
Division Rings

Let V be a countably generated right vector space over a division ring D. If |D| ≠ 2,3, then for any γ ∈ End D(V), there exist α, β ∈ Aut D(V) such that γ-α, γ-α-1, γ2-β2 ∈ Aut D(V).

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