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.

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.

Generalized fine rings

2020 ◽  
pp. 2250060
Author(s):  
Yiqiang Zhou
Keyword(s):  
Jacobson Radical ◽  
Nonzero Element ◽  
Natural Generalization ◽  
Matrix Rings ◽  
New Class

As introduced by Cǎlugǎreanu and Lam in [G. Cǎlugǎreanu and T. Y. Lam, Fine rings: a new class of simple rings, J. Algebra Appl. 15(9) (2016) 1650173, 18 pp.], a fine ring is a ring whose every nonzero element is the sum of a unit and a nilpotent. As a natural generalization of fine rings, a ring is called a generalized fine ring if every element not in the Jacobson radical is the sum of a unit and a nilpotent. Here some known results on fine rings are extended to generalized fine rings. A notable result states that matrix rings over generalized fine rings are generalized fine, extending the important result in [G. Cǎlugǎreanu and T. Y. Lam, Fine rings: a new class of simple rings, J. Algebra Appl. 15(9) (2016) 1650173, 18 pp.] that matrix rings over fine rings are fine.

On semireversible rings

2019 ◽  
pp. 2150018
Author(s):  
Debraj Roy ◽  
Tikaram Subedi
Keyword(s):  
Positive Integer ◽  
Matrix Rings ◽  
New Class ◽  
Ring Of Fractions ◽  
Strongly Regular

In this paper, we introduce and study a new class of rings which we call semireversible rings. A ring [Formula: see text] is called semireversible if for any [Formula: see text] implies there exists a positive integer [Formula: see text] such that [Formula: see text]. We observe that the class of semireversible rings strictly lies between the class of central reversible rings and weakly reversible rings. Some relations are provided between semireversible rings and many other known classes of rings. Some extensions of semireversible rings such as ring of fractions, Dorroh extension, subrings of matrix rings are investigated. Finally, we study semireversible rings via modules over them wherein among other results, we prove that a semireversible left (right) SF-ring is strongly regular.

scholarly journals On distributively generated near-rings

1969 ◽  
Vol 16 (3) ◽  
pp. 239-243 ◽  
Author(s):  
Steve Lich
Keyword(s):  
Division Ring ◽  
Nonzero Element ◽  
Ring Theory ◽  
Left Identity

The following theorems in ring theory are well-known:1. Let R be a ring. If e is a unique left identity, then e is also a right identity.2. If R is a ring with more than one element such that aR = R for every nonzero element a ε R, then R is a division ring.3. A ring R with identity e ≠ 0 is a division ring if and only if it has no proper right ideals.

scholarly journals UU rings

2015 ◽  
Vol 31 (2) ◽  
pp. 157-163
Author(s):  
GRIGORE CALUGAREANU ◽  
Keyword(s):  
Nilpotent Element ◽  
New Class

A new class of rings is studied: rings all whose units are sums 1 + n, for a suitable nilpotent element n. These are called UU rings.

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.

scholarly journals A new class of Hadamard matrices

1967 ◽  
Vol 8 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. Spence
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Identity Matrix ◽  
New Class ◽  
Image Position ◽  
Square Matrix

A Hadamard matrixHis an orthogonal square matrix of ordermall the entries of which are either + 1 or - 1; i. e.whereH′denotes the transpose ofHandImis the identity matrix of orderm. For such a matrix to exist it is necessary [1] thatIt has been conjectured, but not yet proved, that this condition is also sufficient. However, many values ofmhave been found for which a Hadamard matrix of ordermcan be constructed. The following is a list of suchm(pdenotes an odd prime).

High rates of Fibonacci polynomials coding theory

2014 ◽  
Vol 06 (04) ◽  
pp. 1450053
Author(s):  
Bandhu Prasad
Keyword(s):  
Coding Theory ◽  
Fibonacci Polynomials ◽  
New Class ◽  
Matrix Formula ◽  
Fibonacci Polynomial ◽  
Square Matrix

In this paper, a new class of square matrix [Formula: see text] of order pm is introduced, where (p = 3, 4, 5, …), (m = 1, 2, 3, …) and for integers n, x ≥ 1. Fibonacci polynomial coding and decoding methods are followed from [Formula: see text] matrix and high code rates are obtained.

scholarly journals An analogue of Serre’s conjecture for a ring of distributions

2020 ◽  
Vol 8 (1) ◽  
pp. 88-91
Author(s):  
Amol Sasane
Keyword(s):  
Invertible Matrix ◽  
Finitely Generated ◽  
Serre's Conjecture ◽  
Hermite Ring ◽  
Serre’S Conjecture ◽  
Square Matrix

AbstractThe set 𝒜 := 𝔺δ0+ 𝒟+′, obtained by attaching the identity δ0 to the set 𝒟+′ of all distributions on 𝕉 with support contained in (0, ∞), forms an algebra with the operations of addition, convolution, multiplication by complex scalars. It is shown that 𝒜 is a Hermite ring, that is, every finitely generated stably free 𝒜-module is free, or equivalently, every tall left-invertible matrix with entries from 𝒜 can be completed to a square matrix with entries from 𝒜, which is invertible.

On the Generalized Strongly Nil-Clean Property of Matrix Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 625-634
Author(s):  
Aleksandra S. Kostić ◽  
Zoran Z. Petrović ◽  
Zoran S. Pucanović ◽  
Maja Roslavcev
Keyword(s):  
Matrix Rings ◽  
Nilpotent Matrix ◽  
Unital Ring ◽  
Sum Formula ◽  
Matrix Formula

Let [Formula: see text] be an associative unital ring and not necessarily commutative. We analyze conditions under which every [Formula: see text] matrix [Formula: see text] over [Formula: see text] is expressible as a sum [Formula: see text] of (commuting) idempotent matrices [Formula: see text] and a nilpotent matrix [Formula: see text].

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