On feckly polar rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950021
Author(s):  
Tugce Pekacar Calci ◽  
Huanyin Chen
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Structure Theorems ◽  
Regular Rings

In this paper, we introduce a new notion which lies properly between strong [Formula: see text]-regularity and pseudopolarity. A ring [Formula: see text] is feckly polar if for any [Formula: see text] there exists [Formula: see text] such that [Formula: see text] Many structure theorems are proved. Further, we investigate feck polarity for triangular matrix and matrix rings. The relations among strongly [Formula: see text]-regular rings, pseudopolar rings and feckly polar rings are also obtained.

Decompositions into products of idempotents

2015 ◽  
Vol 29 ◽  
pp. 74-88 ◽  
Author(s):  
A. Alahmadi ◽  
S. Jain ◽  
Andre Leroy ◽  
A. Sathaye
Keyword(s):  
Nonnegative Matrix ◽  
Triangular Matrix ◽  
Singular Matrix ◽  
Singular Element ◽  
Property A ◽  
Matrix Rings ◽  
Triangular Matrices ◽  
Rank One ◽  
Nilpotent Matrices ◽  
Regular Rings

The purpose of this note is two-fold: (1) to study when quasi-Euclidean rings, regular rings and regular separative rings have the property (∗) that each right (left) singular element is a product of idempotents, and (2) to consider the question: “when is a singular nonnegative square matrix a product of nonnegative idempotent matrices?” The importance of the class of quasi- Euclidean rings in connection with the property (∗) is given by the first three authors and T.Y. Lam [Journal of Algebra, 406:154–170, 2014], where it is shown that every singular matrix over a right and left quasi-Euclidean domain is a product of idempotents, generalizing the results of J. A Erdos [Glasgow Mathematical Journal, 8: 118–122, 1967] for matrices over fields and that of T. J. Laffey [Linear and Multilinear Algebra, 14:309–314, 1983] for matrices over commutative Euclidean domains. We have shown in this paper that quasi-Euclidean rings appear among many interesting classes of rings and hence they are in abundance. We analyze the properties of triangular matrix rings and upper triangular matrices with respect to the decomposition into product of idempotents and show, in particular, that nonnegative nilpotent matrices are products of nonnegative idempotent matrices. We study as to when each singular matrix is a product of idempotents in special classes of rings. Regarding the second question for nonnegative matrices, bounds are obtained for a rank one nonnegative matrix to be a product of two idempotent matrices. It is shown that every nonnegative matrix of rank one is a product of three nonnegative idempotent matrices. For matrices of higher orders, we show that some power of a group monotone matrix is a product of idempotent matrices.

scholarly journals Functional identities on upper triangular matrix rings

2020 ◽  
Vol 18 (1) ◽  
pp. 182-193
Author(s):  
He Yuan ◽  
Liangyun Chen
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Unital Ring ◽  
Functional Identities ◽  
Upper Triangular Matrix ◽  
Free Subset

Abstract Let R be a subset of a unital ring Q such that 0 ∈ R. Let us fix an element t ∈ Q. If R is a (t; d)-free subset of Q, then Tn(R) is a (t′; d)-free subset of Tn(Q), where t′ ∈ Tn(Q), $\begin{array}{} t_{ll}' \end{array} $ = t, l = 1, 2, …, n, for any n ∈ N.

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

On superderivations and super-biderivations of trivial extensions and triangular matrix rings

2018 ◽  
Vol 47 (4) ◽  
pp. 1662-1670
Author(s):  
Hassan Cheraghpour ◽  
Nader M. Ghosseiri
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings

scholarly journals Gorenstein Projective Modules over Triangular Matrix Rings

2016 ◽  
Vol 23 (01) ◽  
pp. 97-104 ◽  
Author(s):  
H. Eshraghi ◽  
R. Hafezi ◽  
Sh. Salarian ◽  
Z. W. Li
Keyword(s):  
Triangular Matrix ◽  
Projective Modules ◽  
Matrix Rings ◽  
Matrix Algebras ◽  
Gorenstein Projective Modules ◽  
Artin Algebras ◽  
Acyclic Complexes ◽  
Matrix Extension

Let R and S be Artin algebras and Γ be their triangular matrix extension via a bimodule SMR. We study totally acyclic complexes of projective Γ-modules and obtain a complete description of Gorenstein projective Γ-modules. We then construct some examples of Cohen-Macaulay finite and virtually Gorenstein triangular matrix algebras.

Factor-Correspondences in Regular Rings

1982 ◽  
Vol 34 (1) ◽  
pp. 23-30
Author(s):  
S. K. Berberian
Keyword(s):  
Present Article ◽  
Regular Ring ◽  
Decomposition Theorem ◽  
Matrix Rings ◽  
Basic Properties ◽  
Principal Ideals ◽  
Highly Effective ◽  
Left Factor ◽  
Regular Rings

Factor-correspondences are nothing more than a way of describing isomorphisms between principal ideals in a regular ring. However, due to a remarkable decomposition theorem of M. J. Wonenburger [7, Lemma 1], they have proved to be a highly effective tool in the study of completeness properties in matrix rings over regular rings [7, Theorem 1]. Factor-correspondences also figure in the proof of D. Handelman's theorem that an ℵ0-continuous regular ring is unitregular [4, Theorem 3.2].The aim of the present article is to sharpen the main result in [7] and to re-examine its applications to matrix rings. The basic properties of factor-correspondences are reviewed briefly for the reader's convenience.Throughout, R denotes a regular ring (with unity).Definition 1 (cf. [5, p. 209ff], [7, p. 212]). A right factor-correspondence in R is a right R-isomorphism φ : J → K, where J and K are principal right ideals of R (left factor-correspondences are defined dually).

scholarly journals Gorenstein projective modules and recollements over triangular matrix rings

2020 ◽  
Vol 48 (11) ◽  
pp. 4932-4947 ◽  
Author(s):  
Huanhuan Li ◽  
Yuefei Zheng ◽  
Jiangsheng Hu ◽  
Haiyan Zhu
Keyword(s):  
Triangular Matrix ◽  
Projective Modules ◽  
Matrix Rings ◽  
Gorenstein Projective Modules

Differential Inverse Power Series Rings with Quasi-Armendariz-like Condition

2018 ◽  
Vol 25 (04) ◽  
pp. 595-618 ◽  
Author(s):  
Kamal Paykan ◽  
Abasalt Bodaghi
Keyword(s):  
Power Series ◽  
Triangular Matrix ◽  
Inverse Power ◽  
Direct Sums ◽  
Matrix Rings ◽  
Series Ring ◽  
Semiprime Rings ◽  
Armendariz Rings ◽  
Power Series Rings ◽  
Series Type

A generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of differential inverse power series type (or simply, [Formula: see text]-quasi-Armendariz), is introduced and studied. It is shown that the [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. Various classes of non-semiprime [Formula: see text]-quasi-Armendariz rings are provided, and a number of properties of this generalization are established. Some characterizations for the differential inverse power series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP are concluded, where δ is a derivation on the ring R. Finally, miscellaneous examples to illustrate and delimit the theory are given.

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