The structures of a class of $Z$-local rings

Iterated Differential Polynomial Rings over Locally Nilpotent Rings

We study iterated differential polynomial rings over a locally nilpotent ring and show that a large class of such rings are Behrens radical. This extends results of Chebotar and Chen et al.

Weakly left localizable rings

A new class of rings, {\em the class of weakly left localizable rings}, is introduced. A ring $R$ is called {\em weakly left localizable} if each non-nilpotent element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case if a ring has a left Artinian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localiza…

Local loop near-rings

We study loop near-rings, a generalization of near-rings, where the additive structure is not necessarily associative. We introduce local loop near-rings and prove a useful detection principle for localness.

Local and 2-local derivations on Lie matrix rings over commutative involutive rings

In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-adjoint matrices over a commutative $*$-ring is an inner derivation. We also apply our technique to various Lie algebras of infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. We also show that every local spatial derivation on the above Lie algebras is a derivation.

Strong cleanness of the $2\times 2$ matrix ring over a general local ring

A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey \cite{BDD05a} completely characterized the commutative local rings $R$ for which ${\mathbb M}_n(R)$ is strongly clean. For a general local ring $R$ and $n>1$, however, it is unknown when the matrix ring ${\mathbb M}_n(R)$ is strongly clean. Here we completely determine the local rings $R$ for which ${\mathbb M}_2(R)$ is strongly clean.

On weakly locally finite division rings

Weakly locally finite division rings were considered in \cite{dbh}, where it was mentioned that the class of weakly locally finite division rings properly contains the class of locally finite division rings. In this paper, for any integer $n\geq 0$ or $n=\infty$, we construct a weakly locally finite division ring whose Gelfand-Kirillov dimension is $n$. This fact shows in particular that there exist infinitely many weakly locally finite division rings that are not locally finite. Further, for the class of weakly locally finite division rings, we investigate some questions related with the well…

On Knebusch's Norm Principle for quadratic forms over semi-local rings

We prove the version of Knebusch's Norm principle for simple extensions of (semi-)local rings. As an application we prove the Grothedieck-Serre's conjecture on principal homogeneous spaces for the split case of the spinor group.

Square-Free Rings With Local Units

Recently, the author characterized all artinian square-free rings with identity. Here, those results are extended to the setting of rings with local units. We use this characterization of square-free rings to derive many results similar to the artinian case and to determine, in cohomological terms, exactly when a square-free ring has an associated square-free algebra structure.

Hermitian and skew hermitian forms over local rings

We study the classification problem of possibly degenerate hermitian and skew hermitian bilinear forms over local rings where 2 is a unit.

Left localizable rings and their characterizations

A new class of rings, the class of left localizable rings, is introduced. A ring $R$ is left localizable if each nonzero element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a ring to be a left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case if a ring has a left Artinian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets is a left localizable ring iff its left quotient ring is a direct product of finitely many division rings. A c…

On linear groups over weakly locally finite division rings

In this paper, we give the definition of {\em weakly locally finite} division rings and we show that the class of these rings strictly contains the class of locally finite division rings. Further, we study multiplicative subgroups in these rings. Some skew linear groups are also considered. Our new obtained results generalize previous results for centrally finite case.

Block-diagonalization of matrices over local rings.II

Consider rectangular matrices over a local ring R. In the previous work we have obtained criteria for block-diagonalization of such matrices, i.e. U A V=A_1\oplus A_2, where U,V are invertible matrices over R. In this short note we extend the criteria to the decomposability of quiver representations over R.

Commutative Local Rings of bounded module type

It is shown that a local ring R of bounded module type is an almost maximal valuation ring if there exists a non-maximal prime ideal J such that R/J is a maximal valuation domain.

Corings over rings with local units

We show that the category of corings over a fixed base ring with local units is equivalent to the category of comonads in (right) unital modules whose underlying functors preserve inductive limits. Changing base rings, we prove a bi-equivalence of bicategories. A base ring extension of corings by adjunctions is also introduced.

Two-Local derivations on associative and Jordan matrix rings over commutative rings

In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation.

Very clean matrices over local rings

An element $a\in R$ is very clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and either $a-e$ or $a+e$ is invertible. A ring $R$ is very clean in case every element in $R$ is very clean. We explore the necessary and sufficient conditions under which a triangular $2\times 2$ matrix ring over local rings is very clean. The very clean $2\times 2$ matrices over commutative local rings are completely determined. Applications to matrices over power series are also obtained.

Strongly Rad-clean Matrices over Commutative Local Rings

An element $a\in R$ is provided that there exists an idempotent $e\in R$ such that $a-e\in U(R), ae=ea$ and $eae\in J(eRe)$. In this article, we investigate strongly rad-clean matrices over a commutative local ring. We completely determine when a $2\times 2$ matrix over a commutative local ring is strongly rad-clean.

Local morphisms and modules with a semilocal endomorphism ring

An associative ring with 1 is said to be semilocal provided it is semisimple artinian modulo its Jacobson radical, that is, modulo its Jacobson radical it is isomorphic to a finite product of matrices over division rings. Modules with a semilocal endomorphism ring inherit some properties of semisimple modules. For example, they can be decomposed into a finite direct sum of indecomposable submodules, such decomposition is not unique but there are only a finite number of different ones, they cancel from direct sums and they satisfy the n-th root uniqueness property. A ring homomorphism is said t…

2-Local derivations on matrix rings over associative rings

In the present paper it is proved that every inner 2-local derivation on the matrix ring $M_n(\Re)$ of $n\times n$ matrices over a commutative associative ring $\Re$ is an inner derivation. Also, it is proved that, every derivation on an associative ring $\Re$ has an extension to a derivation on the matrix ring $M_n(\Re)$ of $n\times n$ matrices over $\Re$.

Maximal orthogonal sets of unimodular vectors over finite local rings of odd characteristic

Let $R$ be a finite local ring of odd characteristic and $β$ a non-degenerate symmetric bilinear form on $R^2$. In this short note, we determine the largest possible cardinality of pairwise orthogonal sets of unimodular vectors in $R^2$.

2-局部和局部导数在交换反交换环上的乔丹矩阵环

2-Local and local derivations on Jordan matrix rings over commutative involutive rings

研究证明了在交换反环上的自共轭矩阵的乔丹环上，所有2-局部内导数都是导数，并应用于无限维自共轭矩阵值映射的乔丹代数，证明了所有2-局部空间导数都是空间导数。

Type conditions of stable range for identification of qualitative generalized classes of rings

This article deals mostly with the following question: when is the classical ring of quotients of a commutative ring a ring of stable range 1? We introduce the concepts of a ring of (von Neumann) regular range 1, a ring of semihereditary range 1, a ring of regular range 1, a semihereditary local ring, a regular local ring. We find relationships between the introduced classes of rings and known ones, in particular, it is established that a commutative indecomposable almost clean ring is a regular local ring. A commutative any ring of idempotent regular range 1 is an almost clean ring. It is sho…

Multiplicative Subgroups in weakly locally finite division rings

The description of the subgroup structure of a non-commutative division ring is the subject of the intensive study in the theory of division rings in particular, and of the theory of skew linear groups in general. This study is still so far to be complete. In this paper, we study this problem for weakly locally finite division rings. Such division rings constitute a large class which strictly contains the class of locally finite division rings.

局部有限 pro-有限环

Locally finite profinite rings

本文研究局部有限 pro-有限环的结构，分类了半单情形为有限域上完全矩阵环的乘积，并证明其 Jacobson 根幂零，且应用于小紧致 $G$-环的描述。

Automorphisms of Chevalley groups of type $F_4$ over local rings with 1/2

In the paper we prove that every automorphism of a Chevalley group of type $F_4$ over a commutative local ring with~1/2 is standard, i. e., it is a composition of ring and inner automorphisms.

Automorphisms of Chevalley groups of type $B_l$ over local rings with 1/2

In the given paper we prove that every automorphism of a Chevalley group of type $B_l$, $l\geqslant 2$, over a commutative local ring with 1/2 is standard, i. e., it is a composition of ring, inner and central automorphisms.

Dimension theory in iterated local skew power series rings

By analysing the structure of the associated graded ring with respect to certain filtrations, we deduce a number of good properties of iterated local skew power series rings over appropriate base rings. In particular, we calculate the Krull dimension, prime length and global dimension in well-behaved cases, obtaining lower bounds to complement the upper bounds obtained by Venjakob and Wang.

Automorphisms and isomorphisms of Chevalley groups of type $G_2$ over local rings with 1/2 and 1/3

We prove that every isomorphism of Chevalley groups of type $G_2$ over commutative local rings with 1/2 and 1/3 is standard, i. e., it is a composition of a ring isomorphism and an inner automorphism.

Automorphisms of Chevalley groups of types $A_l, D_l, E_l$ over local rings without 1/2

In the given paper we prove that every automorphism of a Chevalley group of type $A_l$, $D_l$, or $E_l$, $l\geqslant 3$, over a commutative local ring without 1/2 is standard, i. e., it is a composition of ring, inner, central and graph automorphisms.

Automorphisms of Chevalley groups of types $A_l, D_l, E_l$ over local rings with 1/2

In the paper we prove that every automorphism of a (elementary) Chevalley group of type $A_l, D_l$, or $E_l$, $l\geqslant 2$, over a commutative local ring with 1/2 is standard, i. e., is the composition of inner, ring, graph and central automorphisms.

Invertible unital bimodules over rings with local units, and related exact sequences of groups

Given an extension $R \subseteq S$ of rings with same set of local units, inspired by the works of Miyashita, we construct four exact sequences of groups relating Picard's groups of $R$ and $S$.

Description of 2-local derivations on some Lie rings of skew-adjoint matrices

In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation.

局部环上一般线性群的普遍等价性

Universal equivalence of general linear groups over local rings with 1/2

研究局部环上一般线性群的普遍等价性，证明其与环的普遍等价性相关联。

A presentation of the symmetric Grothendieck-Witt group of local rings over $\mathbb{F}_2$

Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring $\mathrm{GW}^{\mathrm{s}}(R)$ of $R$ as an abelian group when $R$ has residue field $\mathbb{F}_2$. This completes a recent work by Rogers and Schlichting, where an explicit presentation of $\mathrm{GW}^{\mathrm{s}}(R)$ is given when the residue field is different from $\mathbb{F}_2$. We then use this result to compute the symmetric Grothendieck-Witt rings for the sequences of local rings $\mathbb{Z}/2^n\mathbb{Z}$ and $\mathbb{F}_2[x]/(x^n)$.

Automorphisms of Chevalley groups of types B_2 and G_2 over local rings

In the paper we prove that every automorphism of any adjoint Chevalley group of types B_2 or G_2 is standard, i.e., it is a composition of the ``inner'' automorphism, ring automorphism and central automorphism.

Similarity classes of 3x3 matrices over a local principal ideal ring

In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.

局部环上的弦代数：容许性与双列性

String algebras over local rings: admissibility and biseriality

本文定义局部环上的弦代数，给出容许理想的充分条件，证明其双列性并描述根中单列项的合冲，同时提供离散赋值环上的 Bäckström 序例子。