On Exchange Hermitian Rings

2010 ◽  
Vol 17 (01) ◽  
pp. 87-100 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exchange Ring ◽  
Regular Matrix ◽  
Exchange Rings ◽  
Necessary And Sufficient ◽  
Diagonal Reduction

In this article, we investigate new necessary and sufficient conditions on an exchange ring under which every regular matrix admits a diagonal reduction. We prove that an exchange ring R is an hermitian ring if and only if for any n ≥ 2 and any regular x ∈ Rn, there exists u ∈ CLn(R) such that x = xux; if and only if for any n ≥ 2 and any regular x ∈ Rn, there exists u ∈ CLn(R) such that xu ∈ R is an idempotent. Furthermore, we characterize such exchange rings by means of reflexive inverses and n-pseudo-similarity.

EXCHANGE RINGS OVER WHICH EVERY REGULAR MATRIX ADMITS DIAGONAL REDUCTION

2004 ◽  
Vol 03 (02) ◽  
pp. 207-217 ◽  
Author(s):  
HUANYIN CHEN
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exchange Ring ◽  
Regular Matrix ◽  
Finitely Generated ◽  
Exchange Rings ◽  
Necessary And Sufficient ◽  
Diagonal Reduction

In this paper, we investigate the necessary and sufficient conditions on exchange rings, under which every regular matrix admits diagonal reduction. Also we show that an exchange ring R is strongly separative if and only if for any finitely generated projective right R-module C, if A and B are any right R-modules such that 2C⊕A≅C⊕B, then C⊕A≅B.

On Exchange QB∞-Rings

2007 ◽  
Vol 14 (04) ◽  
pp. 613-623 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exchange Ring ◽  
New Class ◽  
Necessary And Sufficient ◽  
Invertible Matrices ◽  
Diagonal Reduction ◽  
Square Matrix

In this paper, we introduce a new class of rings, the QB∞-rings. We investigate necessary and sufficient conditions under which an exchange ring is a QB∞-ring. The modules over an exchange QB∞-ring are studied. Also, we prove that every regular square matrix over an exchange QB∞-ring admits a diagonal reduction by pseudo-invertible matrices.

scholarly journals On the inclusion of a bounded convergence field in the space of almost convergent sequences

1972 ◽  
Vol 13 (1) ◽  
pp. 82-90 ◽  
Author(s):  
Robert E. Atalla
Keyword(s):  
Sufficient Conditions ◽  
Research Problem ◽  
Necessary And Sufficient Conditions ◽  
General Context ◽  
Regular Matrix ◽  
Reverse Inclusion ◽  
Necessary And Sufficient ◽  
Convergent Sequences

Let T = (tmn) be a regular matrix, and CTbe its bounded convergence field. Necessary and sufficient conditions for CT to contain the space of almost convergent sequences are well known. (See, e.g., [7, p.62]). G. M. Petersen has suggested as a problem for research the discovery of necessary and sufficient conditions for the reverse inclusion: When is CT contained in the space of almost convergent sequences? [7, p. 137, research problem 9]. In this paper we deal with this question in a more general context. First we need some notation.

scholarly journals Some inclusion theorems

1964 ◽  
Vol 6 (4) ◽  
pp. 161-168 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Regular Matrix ◽  
Positive Order ◽  
Riemann Method ◽  
Riesz Method ◽  
The Matrix ◽  
Necessary And Sufficient

1. A number of inclusion theorems have been given in connection with methods of summation which include the Riesz method (R, λ, κ). Lorentz [4, Theorem 10] gives necessary and sufficient conditions for a sequence to sequence regular matrix A = (an, v) to be such that A ⊃ (R, λ, 1)†. He imposes restrictions on the sequence { λn}, so that A does not include all Riesz methods of order 1. In Theorem 1 below, we generalize the Lorentz theorem by giving a condition without restriction on λn, If the matrix A is a series to sequence or series to function regular matrix, there do not appear to be any results concerning the general inclusionA ⊃ (R, λ, κ).However, when A is the Riemann method (ℜ, λ, μ), Russell [7], generalizing earlier results, has given sufficient conditions for (ℜ, λ, μ) ⊃ (R, λ, κ). Our Theorem 2 gives necessary and sufficient conditions for A ⊃ (R, λ, 1), where A satisfies the condition an, v → 1 (n →co, ν fixed). Thus Theorem 2 applies to any series to sequence regular matrix A. In Theorem 3 we give a further representation for matrices A which include (R, λ, 1), and finally make some remarks on the problem of characterizing matrices which include Riesz methods of any positive order κ.

Generalized Stable Exchange Rings

2008 ◽  
Vol 15 (02) ◽  
pp. 193-198 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Exchange Ring ◽  
Regular Matrix ◽  
Exchange Rings ◽  
Invertible Matrices ◽  
Diagonal Reduction ◽  
Stable Ring

A ring R is said to be a generalized stable ring provided that aR + bR = R with a, b ∈ R implies that there exists y ∈ R such that a + by ∈ K(R), where K(R) = {x ∈ R | ∃s, t ∈ R such that sxt = 1}. Let A be a quasi-projective right R-module, and let E = End R(A). If E is an exchange ring, then E is a generalized stable ring if and only if for any R-morphism f : A → M with Im f ≤⊕M and any R-epimorphism g : A → M, there exist e = e2 ∈ E and h ∈ K(E) such that f = g(eh). Furthermore, we prove that every regular matrix over a generalized stable exchange ring admits a diagonal reduction by quasi-invertible matrices.

Finite Exchange Property for Quasi-projective Modules

2011 ◽  
Vol 18 (03) ◽  
pp. 507-518
Author(s):  
Huanyin Chen
Keyword(s):  
Projective Module ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exchange Property ◽  
Projective Modules ◽  
Finitely Generated ◽  
Exchange Rings ◽  
Necessary And Sufficient ◽  
Finitely Generated Modules

In this paper, we obtain several necessary and sufficient conditions under which a quasi-projective module has the finite exchange property. Applications to finitely generated modules are also studied. These extend some corresponding results on exchange rings.

scholarly journals Applications of Hankel and Regular Matrices in Fourier Series

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Abdullah Alotaibi ◽  
M. Mursaleen
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Hankel Matrix ◽  
Necessary And Sufficient Conditions ◽  
Regular Matrix ◽  
Conjugate Fourier Series ◽  
Necessary And Sufficient

Recently, Alghamdi and Mursaleen (2013) used the Hankel matrix to determine the necessary and suffcient condition to find the sum of the Walsh-Fourier series. In this paper, we propose to use the Hankel matrix as well as any general nonnegative regular matrix to obtain the necessary and sufficient conditions to sum the derived Fourier series and conjugate Fourier series.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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