Some Criteria for Hermite Rings and Elementary Divisor Rings

1974 ◽  
Vol 26 (6) ◽  
pp. 1380-1383 ◽  
Author(s):  
Thomas S. Shores ◽  
Roger Wiegand
Keyword(s):  
Triangular Matrix ◽  
Elementary Divisor ◽  
Diagonal Matrix ◽  
Finitely Generated ◽  
Elementary Divisor Ring ◽  
Hermite Ring

Recall that a ring R (all rings considered are commutative with unit) is an elementary divisor ring (respectively, a Hermite ring) provided every matrix over R is equivalent to a diagonal matrix (respectively, a triangular matrix). Thus, every elementary divisor ring is Hermite, and it is easily seen that a Hermite ring is Bezout, that is, finitely generated ideals are principal. Examples have been given [4] to show that neither implication is reversible.

scholarly journals AMALGAMATION OF RINGS DEFINED BY BÉZOUT-LIKE CONDITIONS

2011 ◽  
Vol 10 (06) ◽  
pp. 1343-1350
Author(s):  
MOHAMMED KABBOUR ◽  
NAJIB MAHDOU
Keyword(s):  
Sufficient Conditions ◽  
Elementary Divisor ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Integral Domains ◽  
Bezout Ring ◽  
Elementary Divisor Ring ◽  
Amalgamated Duplication ◽  
Necessary And Sufficient ◽  
Hermite Ring

Let f : A → B be a ring homomorphism and let J be an ideal of B. In this paper, we investigate the transfer of notions elementary divisor ring, Hermite ring and Bézout ring to the amalgamation A ⋈f J. We provide necessary and sufficient conditions for A ⋈f J to be an elementary divisor ring where A and B are integral domains. In this case it is shown that A ⋈f J is an Hermite ring if and only if it is a Bézout ring. In particular, we study the transfer of the previous notions to the amalgamated duplication of a ring A along an A-submodule E of Q(A) such that E2 ⊆ E.

Elementary matrix reduction over Bézout domains

2019 ◽  
Vol 18 (08) ◽  
pp. 1950141
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani Abdolyousefi
Keyword(s):  
Elementary Divisor ◽  
Bezout Domain ◽  
Elementary Matrix ◽  
Matrix Reduction ◽  
Elementary Divisor Ring ◽  
Bezout Domains ◽  
Diagonal Reduction

A ring [Formula: see text] is an elementary divisor ring if every matrix over [Formula: see text] admits a diagonal reduction. If [Formula: see text] is an elementary divisor domain, we prove that [Formula: see text] is a Bézout duo-domain if and only if for any [Formula: see text], [Formula: see text] such that [Formula: see text]. We explore certain stable-like conditions on a Bézout domain under which it is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.

scholarly journals Finite homomorphic images of Bezout duo-domains

2014 ◽  
Vol 6 (2) ◽  
pp. 360-366 ◽  
Author(s):  
O.S. Sorokin
Keyword(s):  
Sufficient Conditions ◽  
Elementary Divisor ◽  
Global Dimension ◽  
Necessary And Sufficient Conditions ◽  
Stable Range ◽  
Bezout Ring ◽  
Injective Modules ◽  
Elementary Divisor Ring ◽  
Necessary And Sufficient ◽  
Free Element

It is proved that for a quasi-duo Bezout ring of stable range 1 the duo-ring condition is equivalent to being an elementary divisor ring. As an application of this result a couple of useful properties are obtained for finite homomorphic images of Bezout duo-domains: they are coherent morphic rings, all injective modules over them are flat, their weak global dimension is either 0 or infinity. Moreover, we introduce the notion of square-free element in noncommutative case and it is shown that they are adequate elements of Bezout duo-domains. In addition, we are going to prove that these elements are elements of almost stable range 1, as well as necessary and sufficient conditions for being square-free element are found in terms of regularity, Jacobson semisimplicity, and boundness of weak global dimension of finite homomorphic images of Bezout duo-domains.

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.

τ-tilting modules over triangular matrix artin algebras

2021 ◽  
pp. 1-23
Author(s):  
Yeyang Peng ◽  
Xin Ma ◽  
Zhaoyong Huang
Keyword(s):  
Matrix Algebra ◽  
Triangular Matrix ◽  
Finitely Generated ◽  
Tilting Modules ◽  
Triangular Matrix Algebra ◽  
Artin Algebras

Let [Formula: see text] and [Formula: see text] be artin algebras and [Formula: see text] the triangular matrix algebra with [Formula: see text] a finitely generated ([Formula: see text])-bimodule. We construct support [Formula: see text]-tilting modules and ([Formula: see text]-)tilting modules in [Formula: see text] from that in [Formula: see text] and [Formula: see text], and give the converse constructions under some condition.

A Hardy-Davies-Petersen Inequality for a Class of Matrices

1978 ◽  
Vol 30 (03) ◽  
pp. 458-465 ◽  
Author(s):  
P. D. Johnson ◽  
R. N. Mohapatra
Keyword(s):  
Triangular Matrix ◽  
Diagonal Matrix ◽  
Lower Triangular Matrix ◽  
Lower Triangular ◽  
Class Of Matrices

Let ω be the set of all real sequences a = ﹛an﹜ n ≧0. Unless otherwise indicated operations on sequences will be coordinatewise. If any component of a has the entry oo the corresponding component of a-1 has entry zero. The convolution of two sequences s and q is given by s * q . The Toeplitz martix associated with sequence s is the lower triangular matrix defined by tnk = sn-k (n ≧ k), tnk = 0 (n < k). It can be seen that Ts(q) = s * q for each sequence q and that Ts is invertible if and only if s0 ≠ 0. We shall denote a diagonal matrix with diagonal sequence s by Ds.

A Stable Range of Class Full Matrices over Elementary Divisor Ring

2014 ◽  
Vol 66 (5) ◽  
pp. 792-795
Author(s):  
B. V. Zabavs’kyi ◽  
B. M. Kuznits’ka
Keyword(s):  
Elementary Divisor ◽  
Stable Range ◽  
Elementary Divisor Ring

On Some n-C2 and Strongly C2 Extensions

2020 ◽  
Vol 27 (03) ◽  
pp. 545-562
Author(s):  
Farid Kourki ◽  
Jianlong Chen ◽  
Wenxi Li
Keyword(s):  
Positive Integer ◽  
Triangular Matrix ◽  
Group Rings ◽  
Finitely Generated ◽  
Matrix Rings ◽  
Left Annihilator

Let R be a ring and n be a positive integer. Then R is called a left n-C2-ring (strongly left C2-ring) if every n-generated (finitely generated) proper right ideal of R has nonzero left annihilator. We discuss some n-C2 and strongly C2 extensions, such as trivial extensions, formal triangular matrix rings, group rings and [Formula: see text][D, C].

scholarly journals Clear rings and clear elements

2021 ◽  
Vol 55 (1) ◽  
pp. 3-9
Author(s):  
B. V. Zabavsky ◽  
O. V. Domsha ◽  
O. M. Romaniv
Keyword(s):  
Regular Element ◽  
Associative Ring ◽  
Elementary Divisor ◽  
Bezout Domain ◽  
Elementary Divisor Ring

An element of a ring $R$ is called clear if it is a sum of a unit-regular element and a unit. An associative ring is clear if each of its elements is clear.In this paper we defined clear rings and extended many results to a wider class. Finally, we proved that a commutative Bezout domain is an elementary divisor ring if and only if every full $2\times 2$ matrix over it is nontrivially clear.

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