Hulls of Ring Extensions

2010 ◽  
Vol 53 (4) ◽  
pp. 587-601 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jae Keol Park ◽  
S. Tariq Rizvi
Keyword(s):  
Group Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Affirmative Answer ◽  
Infinite Matrix ◽  
Matrix Rings ◽  
Morita Equivalent ◽  
Semiprime Rings ◽  
Ring Extensions ◽  
The Right

AbstractWe investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-Baer right ring hulls of R and S, respectively. As an application, we prove that if unital C*-algebras A and B are Morita equivalent as rings, then the bounded central closure of A and that of B are strongly Morita equivalent as C*-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring A[G] of a torsion-free Abelian group G over a commutative semiprime quasi-continuous ring A. Examples that illustrate and delimit the results of this paper are provided.

Projection invariant extending rings

2016 ◽  
Vol 15 (07) ◽  
pp. 1650121 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Adnan Tercan ◽  
Canan C. Yucel
Keyword(s):  
Direct Summand ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Trivial Extension ◽  
Matrix Rings ◽  
Finite Matrix ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
The Right ◽  
Upper Triangular Matrix Ring

A ring [Formula: see text] is said to be right [Formula: see text]-extending if every projection invariant right ideal of [Formula: see text] is essential in a direct summand of [Formula: see text]. In this article, we investigate the transfer of the [Formula: see text]-extending condition between a ring [Formula: see text] and its various ring extensions. More specifically, we characterize the right [Formula: see text]-extending generalized triangular matrix rings; and we show that if [Formula: see text] is [Formula: see text]-extending, then so is [Formula: see text] where [Formula: see text] is an overring of [Formula: see text] which is an essential extension of [Formula: see text], an [Formula: see text] upper triangular matrix ring of [Formula: see text], a column finite or column and row finite matrix ring over [Formula: see text], or a certain type of trivial extension of [Formula: see text].

ANNIHILATOR-STABILITY AND TWO QUESTIONS OF NICHOLSON

2020 ◽  
pp. 1-8
Author(s):  
GUOLI XIA ◽  
YIQIANG ZHOU
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Negative Answer ◽  
Affirmative Answer ◽  
Matrix Rings ◽  
Closed Set ◽  
Ring Form ◽  
Triangular Matrix Ring ◽  
Necessary And Sufficient ◽  
Left Annihilator

Abstract An element a in a ring R is left annihilator-stable (or left AS) if, whenever $Ra+{\rm l}(b)=R$ with $b\in R$ , $a-u\in {\rm l}(b)$ for a unit u in R, and the ring R is a left AS ring if each of its elements is left AS. In this paper, we show that the left AS elements in a ring form a multiplicatively closed set, giving an affirmative answer to a question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.]. This result is used to obtain a necessary and sufficient condition for a formal triangular matrix ring to be left AS. As an application, we provide examples of left AS rings R over which the triangular matrix rings ${\mathbb T}_n(R)$ are not left AS for all $n\ge 2$ . These examples give a negative answer to another question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.] whether R/J(R) being left AS implies that R is left AS.

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.

PURE PROJECTIVITY AND PURE INJECTIVITY OVER FORMAL TRIANGULAR MATRIX RINGS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250107 ◽  
Author(s):  
A. HAGHANY ◽  
M. MAZROOEI ◽  
M. R. VEDADI
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Triangular Matrix Ring

Over a formal triangular matrix ring we study pure injective, pure projective and locally coherent modules. Some applications are then given, in particular the (J-)coherence of the ring [Formula: see text] is characterized whenever BM is flat.

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].

Homological dimensions of special modules over formal triangular matrix rings

2021 ◽  
pp. 2250146
Author(s):  
Lixin Mao
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Special Formula ◽  
Homological Dimensions ◽  
Triangular Matrix Ring

Let [Formula: see text] be a formal triangular matrix ring, where [Formula: see text] and [Formula: see text] are rings and [Formula: see text] is a [Formula: see text]-bimodule. We give some computing formulas of homological dimensions of special [Formula: see text]-modules. As an application, we describe the structures of [Formula: see text]-tilting left [Formula: see text]-modules.

scholarly journals Isomorphism of generalized triangular matrix-rings and recovery of tiles

2003 ◽  
Vol 2003 (9) ◽  
pp. 533-538 ◽  
Author(s):  
R. Khazal ◽  
S. Dăscălescu ◽  
L. Van Wyk
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Commutative Rings ◽  
Local Rings ◽  
Isomorphism Theorem ◽  
Matrix Rings ◽  
Recovery Result

We prove an isomorphism theorem for generalized triangular matrix-rings, over rings having only the idempotents0and1, in particular, over indecomposable commutative rings or over local rings (not necessarily commutative). As a consequence, we obtain a recovery result for the tile in a tiled matrix-ring.

scholarly journals Strongly J-Clean Skew Triangular Matrix Rings

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Yosum Kurtulmaz
Keyword(s):  
Local Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Arbitrary Ring ◽  
Triangular Matrix Ring

Abstract Let R be an arbitrary ring with identity. An element a ∈ R is strongly J-clean if there exist an idempotent e ∈ R and element w ∈ J(R) such that a = e + w and ew = ew. A ring R is strongly J-clean in case every element in R is strongly J-clean. In this note, we investigate the strong J-cleanness of the skew triangular matrix ring Tn(R, σ) over a local ring R, where σ is an endomorphism of R and n = 2, 3, 4.

scholarly journals Relative Gorenstein Dimensions over Triangular Matrix Rings

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2676
Author(s):  
Driss Bennis ◽  
Rachid El Maaouy ◽  
Juan Ramón García Rozas ◽  
Luis Oyonarte
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Projective Dimension ◽  
Global Dimension ◽  
Projective Modules ◽  
Matrix Rings ◽  
Weakly Compatible ◽  
Gorenstein Algebra ◽  
Finite Projective Dimension ◽  
Triangular Matrix Ring

Let A and B be rings, U a (B,A)-bimodule, and T=A0UB the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of GC-projective modules over T. As an application, we study when a morphism in T-Mod is a special GCP(T)-precover and when the class GCP(T) is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension.

Weakly principally quasi-Baer rings

2015 ◽  
Vol 15 (01) ◽  
pp. 1650002 ◽  
Author(s):  
A. Majidinya ◽  
A. Moussavi
Keyword(s):  
Prime Ideal ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Direct Products ◽  
Product Representation ◽  
Triangular Matrix Ring ◽  
Baer Rings ◽  
Upper Triangular Matrix ◽  
The Right ◽  
Minimal Prime

We say a ring R with unity is weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is right s-unital by right semicentral idempotents, which implies that R modulo the right annihilator of any principal right ideal is flat. It is proven that the weakly p.q.-Baer property is inherited by polynomial extensions and includes both left p.q.-Baer rings and right p.q.-Baer rings and is closed under direct products and Morita invariance. A ring R is weakly p.q.-Baer if and only if the upper triangular matrix ring Tn(R) is weakly p.q.-Baer. We extend a theorem of Kist for commutative Baer rings to weakly p.q.-Baer rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. It is also shown that there is an important subclass of weakly p.q.-Baer rings which have a nontrivial subdirect product representation.

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