On 2-nil-good rings

2018 ◽  
Vol 17 (06) ◽  
pp. 1850110
Author(s):  
Marjan Sheibani Abdolyousefi ◽  
Nahid Ashrafi ◽  
Huanyin Chen
Keyword(s):  
Regular Ring ◽  
Matrix Ring ◽  
Good Property ◽  
Fundamental Properties ◽  
Lower Triangular ◽  
Square Matrix

A ring [Formula: see text] is defined to be 2-nil-good if every element in [Formula: see text] is the sum of two units and a nilpotent. Fundamental properties of such rings are obtained. We prove that every strongly [Formula: see text]-regular ring is 2-nil-good if and only if the identity is the sum of two units. One of the main result of this paper is that every square matrix ring over J-fine rings is 2-nil-good. We establish 2-nil-good property for Morita contexts. This implies, in particular, that every matrix ring over 2-nil-good rings is 2-nil-good. Furthermore, we prove that the ring [Formula: see text] of all lower triangular diagonal-finite matrices over a 2-good ring [Formula: see text] is 2-nil-good.

Rings whose cyclics are C3-modules

2016 ◽  
Vol 15 (08) ◽  
pp. 1650152 ◽  
Author(s):  
Yasser Ibrahim ◽  
Xuan Hau Nguyen ◽  
Mohamed F. Yousif ◽  
Yiqiang Zhou
Keyword(s):  
Direct Product ◽  
Regular Ring ◽  
Matrix Ring ◽  
Artinian Ring ◽  
Local Rings ◽  
Classical Theorem ◽  
Basic Properties ◽  
Semiperfect Ring ◽  
Injectivity Condition ◽  
Strongly Regular

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or [Formula: see text]. Here we carry out a study of the rings whose cyclic modules are [Formula: see text]-modules. The motivation is the observation that a ring [Formula: see text] is semisimple artinian if and only if every [Formula: see text] -generated right [Formula: see text]-module is a [Formula: see text]-module. Many basic properties are obtained for the rings whose cyclics are [Formula: see text]-modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a [Formula: see text] matrix ring over a strongly regular ring. Applications to the rings whose [Formula: see text]-generated modules are [Formula: see text] -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed.

Unique Addition Rings

1969 ◽  
Vol 21 ◽  
pp. 1455-1461 ◽  
Author(s):  
W. Stephenson
Keyword(s):  
Binary Operation ◽  
Matrix Ring ◽  
Triangular Matrices ◽  
Ring Isomorphism ◽  
Lower Triangular ◽  
Equivalent Ring ◽  
Lower Triangular Matrices

A semigroup (R, ⋅) is said to be a unique addition ring (UA-ring) if there exists a unique binary operation + making (R, ⋅, + ) into a ring. All our results can be presented in this semigroup theoretic setting. However, we prefer the following equivalent ring theoretic formulation: a ring R is a UA-ring if and only if any semigroup isomorphism α: (R, ⋅) ≅ (S, ⋅) with another ring S is always a ring isomorphism.UA-rings have been studied in (8; 4) and are also touched on in (1; 2; 6; 7). In this note we generalize Rickart's methods to much wider classes of rings. In particular, we show that, for a ring R with a 1 and n ≧ 2, the (n × n) matrix ring over R and its subring of lower triangular matrices are UA-rings.

On reduced rank of triangular matrix rings

2015 ◽  
Vol 14 (04) ◽  
pp. 1550059
Author(s):  
Abigail C. Bailey ◽  
John A. Beachy
Keyword(s):  
Homomorphic Image ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Triangular Matrices ◽  
Reduced Rank ◽  
Artinian Rings ◽  
Triangular Matrix Ring ◽  
Lower Triangular ◽  
Lower Triangular Matrices

We determine conditions under which a generalized triangular matrix ring has finite reduced rank, in the general torsion-theoretic sense. These are applied to characterize certain orders in Artinian rings, and to show that if each homomorphic image of a ring S has finite reduced rank, then so does the ring of lower triangular matrices over S.

scholarly journals A note on rings with the summand sum property

2015 ◽  
Vol 52 (4) ◽  
pp. 450-456
Author(s):  
Shen Liang
Keyword(s):  
Direct Summand ◽  
Regular Ring ◽  
Matrix Ring ◽  
Natural Numbers ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Finite Matrix ◽  
Von Neumann Regular ◽  
Semisimple Ring

A ring R is called right SSP (SIP) if the sum (intersection) of any two direct summands of RR is also a direct summand. Left SSP (SIP) rings are defined similarly. There are several interesting results on rings with SSP. For example, R is right SSP if and only if R is left SSP, and R is a von Neumann regular ring if and only if Mn(R) is SSP for some n > 1. It is shown that R is a semisimple ring if and only if the column finite matrix ring ℂFMℕ(R) is SSP, where ℕ is the set of natural numbers. Some known results are proved in an easy way through idempotents of rings. Moreover, some new results on SSP rings are given.

Metastable Defects in the Amorphous Silicon-Germanium Alloys

2003 ◽  
Vol 762 ◽  
Author(s):  
J. David Cohen
Keyword(s):  
Amorphous Silicon ◽  
Silicon Germanium ◽  
Dangling Bonds ◽  
Annealing Behavior ◽  
Fundamental Properties ◽  
Silicon Materials ◽  
Salient Features ◽  
The Individual ◽  
Amorphous Silicon Germanium ◽  
Silicon Germanium Alloys

AbstractThis paper first briefly reviews a few of the early studies that established some of the salient features of light-induced degradation in a-Si,Ge:H. In particular, I discuss the fact that both Si and Ge metastable dangling bonds are involved. I then review some of the recent studies carried out by members of my laboratory concerning the details of degradation in the low Ge fraction alloys utilizing the modulated photocurrent method to monitor the individual changes in the Si and Ge deep defects. By relating the metastable creation and annealing behavior of these two types of defects, new insights into the fundamental properties of metastable defects have been obtained for amorphous silicon materials in general. I will conclude with a brief discussion of the microscopic mechanisms that may be responsible.

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