Zero Divisor Graphs of Upper Triangular Matrix Rings

2013 ◽  
Vol 41 (12) ◽  
pp. 4622-4636 ◽  
Author(s):  
Aihua Li ◽  
Ralph P. Tucci
Keyword(s):  
Zero Divisor ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Upper Triangular Matrix

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 On p.p.-rings which are reduced

2006 ◽  
Vol 2006 ◽  
pp. 1-5
Author(s):  
Xiaojiang Guo ◽  
K. P. Shum
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Upper Triangular Matrix

Denote the2×2upper triangular matrix rings overℤandℤpbyUTM2(ℤ)andUTM2(ℤp), respectively. We prove that if a ringRis a p.p.-ring, thenRis reduced if and only ifRdoes not contain any subrings isomorphic toUTM2(ℤ)orUTM2(ℤp). Other conditions for a p.p.-ring to be reduced are also given. Our results strengthen and extend the results of Fraser and Nicholson on r.p.p.-rings.

scholarly journals Derivations of upper triangular matrix rings

1993 ◽  
Vol 187 ◽  
pp. 263-267 ◽  
Author(s):  
Sǒnia P. Coelho ◽  
C. Polcino Milies
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Upper Triangular Matrix

Integer-valued polynomials over block matrix algebras

2019 ◽  
Vol 19 (03) ◽  
pp. 2050053
Author(s):  
J. Sedighi Hafshejani ◽  
A. R. Naghipour ◽  
M. R. Rismanchian
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Block Matrix ◽  
Quotient Field ◽  
Matrix Rings ◽  
Matrix Algebras ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Full Matrix Ring ◽  
Upper Triangular Matrix Ring

In this paper, we state a generalization of the ring of integer-valued polynomials over upper triangular matrix rings. The set of integer-valued polynomials over some block matrix rings is studied. In fact, we consider the set of integer-valued polynomials [Formula: see text] for each [Formula: see text], where [Formula: see text] is an integral domain with quotient field [Formula: see text] and [Formula: see text] is a block matrix ring between upper triangular matrix ring [Formula: see text] and full matrix ring [Formula: see text]. In fact, we have [Formula: see text]. It is known that the sets of integer-valued polynomials [Formula: see text] and [Formula: see text] are rings. We state some relations between the rings [Formula: see text] and the partitions of [Formula: see text]. Then, we show that the set [Formula: see text] is a ring for each [Formula: see text]. Further, it is proved that if the ring [Formula: see text] is not Noetherian then the ring [Formula: see text] is not Noetherian, too. Finally, some properties and relations are stated between the rings [Formula: see text], [Formula: see text] and [Formula: see text].

Rings in Which the Annihilator of an Ideal Is Pure

2015 ◽  
Vol 22 (spec01) ◽  
pp. 947-968 ◽  
Author(s):  
A. Majidinya ◽  
A. Moussavi ◽  
K. Paykan
Keyword(s):  
Triangular Matrix ◽  
Complete Characterization ◽  
Direct Products ◽  
Matrix Rings ◽  
Morita Invariant ◽  
Sheaf Representation ◽  
Baer Rings ◽  
Upper Triangular Matrix ◽  
Left Annihilator

A ring R is a left AIP-ring if the left annihilator of any ideal of R is pure as a left ideal. Equivalently, R is a left AIP-ring if R modulo the left annihilator of any ideal is flat. This class of rings includes both right PP-rings and right p.q.-Baer rings (and hence the biregular rings) and is closed under direct products and forming upper triangular matrix rings. It is shown that, unlike the Baer or right PP conditions, the AIP property is inherited by polynomial extensions and has the advantage that it is a Morita invariant property. We also give a complete characterization of a class of AIP-rings which have a sheaf representation. Connections to related classes of rings are investigated and several examples and counterexamples are included to illustrate and delimit the theory.

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