Unitaries in Simple Artinian Rings

1979 ◽  
Vol 31 (3) ◽  
pp. 542-557
Author(s):  
M. Chacron
Keyword(s):  
Division Ring ◽  
Polynomial Identity ◽  
Artinian Ring ◽  
Artinian Rings ◽  
Conjugate Transpose ◽  
Torsion Free

Let R be a 2-torsion free simple artinian ring with involution*. The element u of R is said to be unitary if u is invertible with inverse u*. In this paper we shall be concerned with the subalgebras W of R over its centre Z such that uWu* ⊆ W, for all unitaries u of R. We prove that if R has rank superior to 1 over a division ring D containing more than 5 elements and if R is not 4-dimensional then any such subalgebra W must be one of the trivial subalgebras 0, Z or R, under one of the following extra finiteness assumptions: W contains inverses, W satisfies a polynomial identity, the ground division ring D is algebraic, the involution is a conjugate-transpose involution such that D equipped with the induced involution is generated by unitaries.

Homological Aspects of Semigroup Gradings on Rings and Algebras

1999 ◽  
Vol 51 (3) ◽  
pp. 488-505 ◽  
Author(s):  
W. D. Burgess ◽  
Manuel Saorín
Keyword(s):  
Euler Characteristic ◽  
Division Ring ◽  
Artinian Ring ◽  
Global Dimension ◽  
Monomial Algebra ◽  
Finite Dimensional ◽  
Cartan Matrices ◽  
Torsion Free ◽  
Rings And Algebras ◽  
Finite Dimensional Algebras

AbstractThis article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup Σ. Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the Σ-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties.A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert Σ-series in the associated path incidence ring.The rationality of the Σ-Euler characteristic, the Hilbert Σ-series and the Poincaré-Betti Σ-series is studied when Σ is torsion-free commutative and A is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.

Rings whose Indecomposable Injective Modules are Uniserial

1982 ◽  
Vol 34 (4) ◽  
pp. 797-805 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Artinian Ring ◽  
Dimensional Algebra ◽  
Artinian Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Finite Dimensional ◽  
Uniserial Modules ◽  
Left And Right ◽  
Finitely Generated Modules ◽  
Algebra Over A Field

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

Fine rings: A new class of simple rings

2016 ◽  
Vol 15 (09) ◽  
pp. 1650173 ◽  
Author(s):  
G. Cǎlugǎreanu ◽  
T. Y. Lam
Keyword(s):  
Division Ring ◽  
Nilpotent Element ◽  
Nonzero Element ◽  
Proper Class ◽  
Invertible Matrix ◽  
Matrix Rings ◽  
Artinian Rings ◽  
New Class ◽  
Nilpotent Matrix ◽  
Square Matrix

A nonzero ring is said to be fine if every nonzero element in it is a sum of a unit and a nilpotent element. We show that fine rings form a proper class of simple rings, and they include properly the class of all simple artinian rings. One of the main results in this paper is that matrix rings over fine rings are always fine rings. This implies, in particular, that any nonzero (square) matrix over a division ring is the sum of an invertible matrix and a nilpotent matrix.

Hereditary artinian rings of finite representation type and extensions of simple artinian rings

1987 ◽  
Vol 102 (3) ◽  
pp. 411-420 ◽  
Author(s):  
Aidan Schofield
Keyword(s):  
Artinian Ring ◽  
Representation Type ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Artinian Rings ◽  
Finite Dimensional ◽  
Coxeter Diagram ◽  
Skew Fields ◽  
Left And Right ◽  
Natural Way

In [1], Dowbor, Ringel and Simson consider hereditary artinian rings of finite representation type; it is known that if A is an hereditary artinian algebra of finite representation type, finite-dimensional over a field, then it corresponds to a Dynkin diagram in a natural way; they show that an hereditary artinian ring of finite representation type corresponds to a Coxeter diagram. However, in order to construct an hereditary artinian ring of finite representation type corresponding to a Coxeter diagram that is not Dynkin, they show that it is necessary though not sufficient to find an extension of skew fields such that the left and right dimensions are both finite but are different. No examples of such skew fields were known at the time. In [3], I constructed such extensions, and the main aim of this paper is to extend the methods of that paper to construct an extension of skew fields having all the properties needed to construct an hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5).

scholarly journals Left localizations of left Artinian rings

2016 ◽  
Vol 15 (09) ◽  
pp. 1650165 ◽  
Author(s):  
V. V. Bavula
Keyword(s):  
Artinian Ring ◽  
Isomorphism Classes ◽  
Artinian Rings

For an arbitrary left Artinian ring [Formula: see text], explicit descriptions are given of all the left denominator sets [Formula: see text] of [Formula: see text] and left localizations [Formula: see text] of [Formula: see text]. It is proved that, up to [Formula: see text]-isomorphism, there are only finitely many left localizations and each of them is an idempotent localization, i.e. [Formula: see text] and [Formula: see text] where [Formula: see text] is a left denominator set of [Formula: see text] and [Formula: see text] is an idempotent. Moreover, the idempotent [Formula: see text] is unique up to a conjugation. It is proved that the number of maximal left denominator sets of [Formula: see text] is finite and does not exceed the number of isomorphism classes of simple left [Formula: see text]-modules. The set of maximal left denominator sets of [Formula: see text] and the left localization radical of [Formula: see text] are described.

A Note on Compactifying Artinian Rings

1974 ◽  
Vol 26 (3) ◽  
pp. 580-582 ◽  
Author(s):  
David K. Haley
Keyword(s):  
Finite Group ◽  
Artinian Ring ◽  
Topological Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
A Finite Group

In this note a number of compactifications are discussed within the class of artinian rings. In [1] the following was proved:Theorem. For an artinian ring R the following are equivalent:(1) R is equationally compact.(2) R+ ≃ B ⊕ P, where B is a finite group, P is a finite direct sum of Prüfer groups, and R · P = P · R = {0}.(3) R is a retract of a compact topological ring.

Direct Sums of Torsion-Free Covers

1973 ◽  
Vol 25 (5) ◽  
pp. 1002-1005
Author(s):  
Thomas Cheatham
Keyword(s):  
Direct Product ◽  
Commutative Rings ◽  
Integral Domains ◽  
Direct Sums ◽  
Artinian Rings ◽  
Direct Sum ◽  
Special Case ◽  
Torsion Free

In [4, Theorem 4.1, p. 45], Enochs characterizes the integral domains with the property that the direct product of any family of torsion-free covers is a torsion-free cover. In a setting which includes integral domains as a special case, we consider the corresponding question for direct sums. We use the notion of torsion introduced by Goldie [5]. Among commutative rings, we show that the property “any direct sum of torsion-free covers is a torsion-free cover“ characterizes the semi-simple Artinian rings.

scholarly journals Rigid Artinian rings

1982 ◽  
Vol 25 (1) ◽  
pp. 97-99 ◽  
Author(s):  
K. R. McLean
Keyword(s):  
Cyclic Group ◽  
Characteristic Zero ◽  
Prime Power ◽  
Artinian Ring ◽  
Ring Of Integers ◽  
Artinian Rings

In [4], Maxson studied the properties of a ring R whose only ring endomorphisms φ: R → R are the trivial ones, namely the identity map, idR, and the map 0R given by φ(R) = 0. We shall say that any such ring is rigid, slightly extending the definition used in [4] by dropping the restriction that R2 ≠ 0. Maxson's most detailed results concerned the structure of rigid artinian rings, and our main aim is to complete this part of his investigation by establishing the followingTheorem. Let R(≠0) be a left-artinian ring. Then R is rigid if and only if(i) , the ring of integers modulo a prime power pk,(ii) R ≅ N2, the null ring on a cyclic group of order 2, or(iii) R is a rigid field of characteristic zero.

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.

A CONSTRUCTION OF RIGHT ARTINIAN RIGHT SERIAL RINGS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350057
Author(s):  
SURJEET SINGH
Keyword(s):  
Homomorphic Image ◽  
Artinian Ring ◽  
Finitely Generated ◽  
Serial Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
Non Local ◽  
Universal Construction ◽  
Generating System

A ring R is said to be right serial, if it is a direct sum of right ideals which are uniserial. A ring that is right serial need not be left serial. Right artinian, right serial ring naturally arise in the study of artinian rings satisfying certain conditions. For example, if an artinian ring R is such that all finitely generated indecomposable right R-modules are uniform or all finitely generated indecomposable left R-modules are local, then R is right serial. Such rings have been studied by many authors including Ivanov, Singh and Bleehed, and Tachikawa. In this paper, a universal construction of a class of indecomposable, non-local, basic, right artinian, right serial rings is given. The construction depends on a right artinian, right serial ring generating system X, which gives rise to a tensor ring T(L). It is proved that any basic right artinian, right serial ring is a homomorphic image of one such T(L).

Sign in / Sign up

Export Citation Format

Share Document