UN-rings

2016 ◽  
Vol 15 (10) ◽  
pp. 1650182 ◽  
Author(s):  
Grigore Călugăreanu
Keyword(s):  
Division Ring ◽  
Nilpotent Element ◽  
Proper Class ◽  
Invertible Matrix ◽  
Artinian Rings ◽  
Nilpotent Matrix ◽  
Unit Form

A nonzero ring is called a UN-ring if every nonunit is a product of a unit and a nilpotent element. We show that all simple Artinian rings are UN-rings and that the UN-rings whose identity is a sum of two units (e.g. if 2 is a unit), form a proper class of 2-good rings (in the sense of P. Vámos). Thus, any noninvertible matrix over a division ring is the product of an invertible matrix and a nilpotent matrix.

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.

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.

A Trace Condition Equivalent to Simultaneous Triangularizability

1986 ◽  
Vol 38 (2) ◽  
pp. 376-386 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Division Ring ◽  
Sufficient Conditions ◽  
Invertible Matrix ◽  
Multiplicative Semigroup ◽  
Nilpotent Matrices

A collection of matrices over a field F is said to be triangularizable if there is an invertible matrix T over F such that the matrices T−1ST, are all upper triangular. It is a well-known and easy fact that any commutative set is triangularizable if F is algebraically closed, or if F contains the spectrum of every member of . Many sufficient conditions are known for triangularizability of matrix collections. Levitzki [7] proved that a (multiplicative) semigroup of nilpotent matrices is triangularizable. (His result is valid even over a division ring.) Kolchin [5] showed the triangularizability of a semigroup of unipotent matrices, i.e., matrices of the form I + N with N nilpotent. Kaplansky [3, 4] unified and generalized these results.

scholarly journals Artinian bimodule with quasi-Frobenius bimodule of translations

2019 ◽  
Vol 29 (2) ◽  
pp. 103-119
Author(s):  
Aleksandr A. Nechaev ◽  
Vadim N. Tsypyschev
Keyword(s):  
Commutative Ring ◽  
Recurrent Sequence ◽  
Artinian Rings ◽  
Linear Recurrent Sequence ◽  
Left And Right ◽  
Additional Assumptions

Abstract The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule AMB over left- and right-Artinian rings A and B, respectively, is associated with the equivalent bimodule of translations CMZ, where C is the multiplicative ring of the bimodule AMB and Z is its center, and the relation between the quasi-Frobenius conditions for the bimodules AMB and CMZ is studied. It is demonstrated that, in the general case, the fact that AMB is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule CMZ. However, under some additional assumptions it can be shown that if CMZ is a quasi-Frobenius bimodule, then the bimodule AMB is quasi-Frobenius as well.

scholarly journals Design of PDC Controllers by Matrix Reversibility for Synchronization ofYinandYangChaotic Takagi-Sugeno Fuzzy Henon Maps

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Chun-Yen Ho ◽  
Hsien-Keng Chen ◽  
Zheng-Ming Ge
Keyword(s):  
Chaos Synchronization ◽  
Matrix Theory ◽  
Chinese Philosophy ◽  
Fuzzy Model ◽  
Invertible Matrix ◽  
Henon Maps ◽  
Hénon Maps ◽  
Feminine Principle ◽  
Simulation Results ◽  
Takagi Sugeno

This paper investigates the synchronization ofYinandYangchaotic T-S fuzzy Henon maps via PDC controllers. Based on the Chinese philosophy,Yinis the decreasing, negative, historical, or feminine principle in nature, whileYangis the increasing, positive, contemporary, or masculine principle in nature.YinandYangare two fundamental opposites in Chinese philosophy. The Henon map is an invertible map; so the Henon maps with increasing and decreasing argument can be called theYangandYinHenon maps, respectively. Chaos synchronization ofYinandYangT-S fuzzy Henon maps is achieved by PDC controllers. The design of PDC controllers is based on the linear invertible matrix theory. The T-S fuzzy model ofYinandYangHenon maps and the design of PDC controllers are novel, and the simulation results show that the approach is effective.

A GENERALIZATION OF SEMISIMPLE ARTINIAN RINGS

2005 ◽  
Vol 04 (03) ◽  
pp. 231-235
Author(s):  
YASUYUKI HIRANO ◽  
HISAYA TSUTSUI
Keyword(s):  
Direct Summand ◽  
Artinian Rings

We investigate a ring R with the property that for every right R-module M and every ideal I of R the annihilator of I in M is a direct summand of M, and determine conditions under which such a ring is semisimple Artinian.

On the Lie Ring of a Division Ring

1954 ◽  
Vol 60 (3) ◽  
pp. 571 ◽  
Author(s):  
I. N. Herstein
Keyword(s):  
Division Ring ◽  
Lie Ring

On commutative rings with uniserial dimension

2014 ◽  
Vol 14 (01) ◽  
pp. 1550008 ◽  
Author(s):  
A. Ghorbani ◽  
Z. Nazemian
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Artinian Rings ◽  
Uniform Dimension ◽  
Uniserial Modules

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

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