Non-commutative Local Rings Research Articles

Type IV codes over a non-unital ring

There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: see text] and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

On maximal commutative subrings of non-commutative rings

ABSTRACTWe observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.

Quasipolar Property of Generalized Matrix Rings

The article concerns the question of when a generalized matrix ring K s (R) over a local ring R is quasipolar. For a commutative local ring R, it is proved that K s (R) is quasipolar if and only if it is strongly clean. For a general local ring R, some partial answers to the question are obtained. There exist noncommutative local rings R such that K s (R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix of K s (R) (where R is a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5] are improved or extended.

FINITE LOCAL RINGS OF ORDER ≤ 16 WITH NONZERO JACOBSON RADICAL

The structures of finite local rings of order <TEX>${\leq}$</TEX> 16 with nonzero Jacobson radical are investigated. The whole shape of non-commutative local rings of minimal order is completely determined up to isomorphism.

Strongly J-Clean Matrices Over Local Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8, Theorems 2.7 and 3.2], [9, Theorem 2.6], and [11, Theorem 7].

Algebraicity of the radical in local rings

In this paper we continue the study of algebraic subsets of non-commutative local rings. (A subset of a ring is said to be algebraic if there exists a monic polynomial with coefficients from the ring vanishing on the subset.) In particular, we prove that the Jacobson radical of a local ring is an algebraic subset if and only if it is a nil ideal of bounded index.

On local right pure semisimple rings of length two or three

We investigate in this paper non-commutative local rings of the smallest length that are potential counter-examples to the pure semisimplicity conjecture. Throughout the paper is an associative ring with an identity element. We call local, if the Jacobson radical ( ) of is a two-sided maximal ideal. We denote by mod( ) the category of finitely generated right -modules. Given a right -module of finite length we denote by ( ) the length of . We recall that a ring is said to be of finite representation type, if is artinian and the number of the isomorphism classes of finitely generated indecomposable right (and left) -modules is finite. Following [24] we call a ring right pure semisimple, if every right -module is a direct sum of finitely generated modules. It is well known that a ring is of finite representation type if and only if is right pure semisimple and is left pure semisimple (see [2], [11], [18], [22]–[24]). It is still an open question, called the pure semisimplicity conjecture, if a right pure semisimple ring is of finite representation type (see [2] and [24], [25], [28]). In [13] the question is answered in affirmative for rings satisfying a polynomial identity and for self-injective rings (see also [7], [19] and [31]). The reader is referred to [42] and to the author’s expository papers [30] and [32] for a basic background and historical comments on the pure semisimplicity conjecture. It was shown by the author in [28] and [33] that there is a chance to find a counter-example to the pure semisimplicity conjecture and might be hereditary with two simple non-isomorphic modules. The existence of a counter-example depends on a generalized Artin problem on division ring extensions. In the present paper we are mainly interested in the existence of counter-examples to pure semisimplicity conjecture that are local of the smallest length, that is, of length ( ) two or three. This continues our study started in [28], [35] and [33]. It is shown in Lemma 3.1 that every such a local ring has ( )2 = 0. Therefore we study representation-infinite right pure semisimple local rings with ( )2 = 0 such that the Auslander-Reiten quiver (mod ) is of the form · · · → •→ •→ •→

Dualizing Complexes over Noncommutative Local Rings

We prove an existence theorem for dualizing complexes over noncommutative noetherian complete semilocal algebras, which generalizes Van den Bergh's existence theorem in the graded case. Using the dualizing complex, noncommutative versions of Bass theorem and the no-holes theorem are proved. We also prove that noetherian complete semilocal algebras satisfying polynomial identities are catenary.

Frobenius type theorems in the noncommutative case *

The classification of semilinear transformations preserving the determinant over a noncommutative local ring is given. For local Ore domains we obtain a classification of Adjamagbo determinant preservers. The developed technique allows us to classify Dieudonné determinant preservers for division rings which may be infinite dimensional over the center.

Singularity preservers over local domains

The classification of semilinear singularity preservers for matrices over noncommutative local rings is given.

Heisenberg algebras and coefficient rings*

In the noncommutative theory of local rings, the existence of coefficient fields is not always granted. We study a counter-example constructed using an enveloping algebra of a Heisenberg algebra to see how to describe a good coeficient ring for a non-commutative local ring, with commutative residue field. Fhrther, dealing with the case of a noncommutative residue division algebra, we use a theorem of Hochschild on the Brauer group to describe a canonical coefficient ring in the case when the exponent of the residue division algebra is prime to the characteristic of the residue field.*

Nonabelian tensor products and nonabelian homology of groups

The non-abelian tensor product of groups introduced by Brown and Loday is generalized. The homology groups of groups are constructed with coefficients in any group, as the left derived functors of the non-abelian tensor product, which generalize the classical theory of homology of groups. Exact sequences of the non-abelian homology groups and their application to algebraic K-theory of noncommutative local rings are given.

On subgroups ofGL 2 over non-commutative local rings which are normalized by elementary matrices

On subgroups ofGL 2 over non-commutative local rings which are normalized by elementary matrices

Locally Archimedian right-ordered groups and locally invariant valuation rings

In the last years there has been a rapid development in noncommutative valuation theory, in particular pursued by Mathiak in [ 11, 12, 131. His approach overcomes the difficulties that the (generalized) value groups are no longer linearly ordered as well as the associated valuation rings have “only a few” two-sided ideals. From the point of view of ring theory valuation rings (in the sense of Mathiak) are chain rings (without zero divisors), namely rings whose lattices of right and left ideals are linearly ordered. There are numerous articles answering ring-theoretical questions on that class of noncommutative local rings. Although Mathiak’s approach is (in some sense) as general as possible mainly those “strongly” noncommutative valuations are of interest which can still be treated; e.g., a modified approximation theorem for these valuations is valid. It was Grater [9] who introduced a special class of noncommutative valuations, namely the so called “locally invariant valuations.” In Section 1 of this article we investigate a class of right chain rings which contains the valuation rings of Grater’s locally invariant valuations. However, our definition is more general including right chain rings which are not necessarily chain rings and admitting zero-divisors. It turns out, that these rings can be characterized by the existence of not too many, but “sufficiently many” two-sided ideals, to be more precise: between any two prime ideals there is a further two-sided ideal (see Proposition 1.3). As the multiplicative semigroups of these rings are closely related to certain right ordered groups, Section 2 is devoted to a new characterization of a class of right-ordered groups (Theorem 2.8) called locally archimedian (or sometimes poly-ordered) groups, which has earlier been studied by Conrad [S]. A positive answer to a question of Ault [2] is given. In Section 3 we prove that there is a close connection between the locally invariant chain domains introduced by Grater and these right-ordered 328 0021-8693/87 $3.00

Correction to: Fundamental theorem for the projective line over non-commutative local rings

Correction to: Fundamental theorem for the projective line over non-commutative local rings

Fundamental theorem for the projective line over non-commutative local rings

Fundamental theorem for the projective line over non-commutative local rings

Some Remarks on Non-Commutative Extensions of Local Rings

In [1] we introduced the concept of a non-commutative local ring and studied the structure of such rings. Unfortunately, we were not able to show that the completion of a local ring was a semi-local ring. In this paper we propose to study a class of rings for which the above result is valid. This class of rings is the integral extensions -[4, 5]-of commutative local rings. This class of rings includes the important class of matrix rings over commutative local rings. In part 1 below we study some elementary properties of integral extensions and here we assume merely that the underlying ring is semi-local. In part 2 we discuss some questions of ideal theory for arbitrary local rings as well as for integral extensions. In a later paper we propose to utilize our results to study the deeper properties of these rings including a dimension theory for such rings. We are particularly indebted to the work of Nagata [6, 7, 8, 9] in the preparation of this paper.