Rings for which every cyclic module is dual automorphism-invariant

2016 ◽  
Vol 15 (05) ◽  
pp. 1650078 ◽  
Author(s):  
M. Tamer Koşan ◽  
Nguyen Thi Thu Ha ◽  
Truong Cong Quynh
Keyword(s):  
Matrix Ring ◽  
Cyclic Module ◽  
The Matrix ◽  
Semiperfect Ring

Rings all of whose right ideals are automorphism-invariant are called right [Formula: see text]-rings. In the present paper, we study rings having the property that every right cyclic module is dual automorphism-invariant. Such rings are called right [Formula: see text]-rings. We obtain some of the relationships [Formula: see text]-rings and [Formula: see text]-rings. We also prove that; (i) A semiperfect ring [Formula: see text] is a right [Formula: see text]-ring if and only if any right ideal in [Formula: see text] is a left [Formula: see text]-module, where [Formula: see text] is a subring of [Formula: see text] generated by its units, (ii) [Formula: see text] is semisimple artinian if and only if [Formula: see text] is semiperfect and the matrix ring [Formula: see text] is a right [Formula: see text]-ring for all [Formula: see text], (iii) Quasi-Frobenius right [Formula: see text]-rings are Frobenius.

scholarly journals On two open problems about strongly clean rings

2004 ◽  
Vol 70 (2) ◽  
pp. 279-282 ◽  
Author(s):  
Zhou Wang ◽  
Jianlong Chen
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Open Problems ◽  
Clean Rings ◽  
Clean Ring ◽  
The Matrix ◽  
Triangular Matrix Ring ◽  
Semiperfect Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. In 1999 Nicholson asked whether every semiperfect ring is strongly clean and whether the matrix ring of a strongly clean ring is strongly clean. In this paper, we prove that if R = {m/n ∈ ℚ: n is odd}, then M2(R) is a semiperfect ring but not strongly clean. Thus, we give negative answers to both questions. It is also proved that every upper triangular matrix ring over the ring R is strongly clean.

Wear Model for Piston Ring and Cylinder Bore System

2005 ◽  
Author(s):  
Yunchao Qiu ◽  
Qian Zou ◽  
Gary C. Barber ◽  
Harold E. McCormick ◽  
Dequan Zou ◽  
...  
Keyword(s):  
Piston Ring ◽  
Thickness Distribution ◽  
Matrix Ring ◽  
Experimental Investigations ◽  
General Tendency ◽  
Wear Model ◽  
Wear Process ◽  
The Matrix ◽  
Surface Profiles ◽  
Cylinder Bore

A new wear model for piston ring and cylinder bore system has been developed to predict wear process with high accuracy and efficiency. It will save time and cost compared with experimental investigations. Surfaces of ring and bore were divided into small domains and assigned to corresponding elements in two-dimensional matrix. Fast Fourier Transform (FFT) and Conjugate Gradient Method (CGM) were applied to obtain pressure distribution on the computing domain. The pressure and film thickness distribution were provided by a previously developed ring/bore lubrication module. By changing the wear coefficients of the ring and bore with accumulated cycles, wear was calculated point by point in the matrix. Ring and bore surface profiles were modified when wear occurred. The results of ring and bore wear after 1 cycle, 10 cycles and 2 hours at 3600 rpm were calculated. They coincided well with the general tendency of wear in a ring and bore system.

scholarly journals Complementation of the Jacobson group in a matrix ring

2005 ◽  
Vol 72 (2) ◽  
pp. 317-324
Author(s):  
David Dolžan
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Matrix Ring ◽  
Graded Rings ◽  
Matrix Rings ◽  
Generators And Relations ◽  
Group Of Units ◽  
The Matrix ◽  
Finite Commutative Ring

The Jacobson group of a ring R (denoted by  = (R)) is the normal subgroup of the group of units of R (denoted by G(R)) obtained by adding 1 to the Jacobson radical of R (J(R)). Coleman and Easdown in 2000 showed that the Jacobson group is complemented in the group of units of any finite commutative ring and also in the group of units a n × n matrix ring over integers modulo ps, when n = 2 and p = 2, 3, but it is not complemented when p ≥ 5. In 2004 Wilcox showed that the answer is positive also for n = 3 and p = 2, and negative in all the remaining cases. In this paper we offer a different proof for Wilcox's results and also generalise the results to a matrix ring over an arbitrary finite commutative ring. We show this by studying the generators and relations that define a matrix ring over a field. We then proceed to examine the complementation of the Jacobson group in the matrix rings over graded rings and prove that complementation depends only on the 0-th grade.

scholarly journals Non-associative algebras containing the matrix ring

2008 ◽  
Vol 429 (1) ◽  
pp. 72-78 ◽  
Author(s):  
Jongwoo Lee ◽  
Ki-Bong Nam
Keyword(s):  
Matrix Ring ◽  
Associative Algebras ◽  
The Matrix

Derivations of some classes of matrix rings

2017 ◽  
Vol 16 (02) ◽  
pp. 1750027 ◽  
Author(s):  
Feride Kuzucuoğlu ◽  
Umut Sayın
Keyword(s):  
Associative Ring ◽  
Matrix Ring ◽  
Matrix Rings ◽  
Ideal Formula ◽  
The Matrix

Let [Formula: see text] be the ring of all (lower) niltriangular [Formula: see text] matrices over any associative ring [Formula: see text] with identity and [Formula: see text] be the ring of all [Formula: see text] matrices over an ideal [Formula: see text] of [Formula: see text]. We describe all derivations of the matrix ring [Formula: see text].

Algebraic elements in matrix ring over division algebras

1995 ◽  
Vol 118 (2) ◽  
pp. 215-221
Author(s):  
A. I. Lichtman
Keyword(s):  
Finite Group ◽  
Matrix Ring ◽  
Quotient Ring ◽  
Arbitrary Field ◽  
Division Algebras ◽  
Nilpotent Radical ◽  
Ring Of Fractions ◽  
The Matrix ◽  
Algebraic Elements ◽  
Linear Envelope

Let K be an arbitrary field, G a polycyclic-by-finite group and A a prime ideal of the group ring KG. It is well known that the quotient ring (KG)/A is a Goldie ring; we denote by R its ring of fractions. Let U be a subgroup of units of the matrix ring Rn×n let K[U] be the linear envelope of U and let rad (K[U]) be the nilpotent radical of K [U].

ON THE CARDINALITY OF SUBSETS OF THE MATRIX RING OVER CERTAIN RESIDUE RING

2018 ◽  
Vol 40 (6) ◽  
pp. 1079-1087
Author(s):  
Fumitsuna Maruyama ◽  
Yozo Deguchi ◽  
Masao Toyoizumi
Keyword(s):  
Matrix Ring ◽  
Residue Ring ◽  
The Matrix

A nil-clean 2 × 2 matrix over the integers which is not clean

2014 ◽  
Vol 13 (06) ◽  
pp. 1450009 ◽  
Author(s):  
Dorin Andrica ◽  
Grigore Călugăreanu
Keyword(s):  
Matrix Ring ◽  
Clean Ring ◽  
The Matrix ◽  
Nil Clean Ring ◽  
Clean Element

While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring ℳ2(Z) which is not clean.

scholarly journals On power sums of matrices over a finite commutative ring

2017 ◽  
Vol 27 (05) ◽  
pp. 547-560 ◽  
Author(s):  
P. Fortuny ◽  
J. M. Grau ◽  
A. M. Oller-Marcén ◽  
I. F. Rúa
Keyword(s):  
Commutative Ring ◽  
Matrix Ring ◽  
Power Sums ◽  
The Matrix ◽  
Finite Commutative Ring

In this paper, we deal with the problem of computing the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] with [Formula: see text] and [Formula: see text] a finite commutative ring. We completely solve the problem in the case [Formula: see text] and give some results that compute the value of this sum if [Formula: see text] is an arbitrary finite commutative ring for many values of [Formula: see text] and [Formula: see text]. Finally, based on computational evidence and using some technical results proved in this paper, we conjecture that the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] is always [Formula: see text] unless [Formula: see text], [Formula: see text], [Formula: see text] and the only element [Formula: see text] such that [Formula: see text] is idempotent, in which case the sum is [Formula: see text].

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