Serial Right Noetherian Rings

1984 ◽  
Vol 36 (1) ◽  
pp. 22-37 ◽  
Author(s):  
Surjeet Singh
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Discrete Valuation Ring ◽  
Noetherian Rings ◽  
Direct Sums ◽  
Matrix Rings ◽  
Serial Ring ◽  
The Family ◽  
Triangular Matrix Ring ◽  
Right Noetherian Rings

A module M is called a serial module if the family of its submodules is linearly ordered under inclusion. A ring R is said to be serial if RR as well as RR are finite direct sums of serial modules. Nakayama [8] started the study of artinian serial rings, and he called them generalized uniserial rings. Murase [5, 6, 7] proved a number of structure theorems on generalized uniserial rings, and he described most of them in terms of quasi-matrix rings over division rings. Warfield [12] studied serial both sided noetherian rings, and showed that any such indecomposable ring is either artinian or prime. He further showed that a both sided noetherian prime serial ring is an (R:J)-block upper triangular matrix ring, where R is a discrete valuation ring with Jacobson radical J. In this paper we determine the structure of serial right noetherian rings (Theorem 2.11).

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.

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].

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.

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.

scholarly journals Generalized blocked triangular matrix rings associated with finite abelian centralizer near-rings

1998 ◽  
Vol 41 (1) ◽  
pp. 177-195
Author(s):  
Kirby C. Smith ◽  
Leon Van Wyk
Keyword(s):  
Large Class ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Abelian Centralizer ◽  
Triangular Matrix Ring ◽  
Structural Matrix Ring ◽  
Structural Matrix

For N any member of a large class of finite abelian right centralizer near-rings, the subring of the ring End(N) of endomorphisms of (N, +) generated by the set of right multiplication maps on N is explicitly described as a generalized blocked triangular matrix ring, which in some cases turns out to be a structural matrix ring.

Nonsingular rings with essential socles

1974 ◽  
Vol 17 (3) ◽  
pp. 358-375 ◽  
Author(s):  
G. Ivanov
Keyword(s):  
Matrix Representation ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Primitive Idempotent ◽  
Matrix Rings ◽  
Uniserial Ring ◽  
Special Cases ◽  
Triangular Matrix Ring ◽  
General Method

This paper is a study of nonsingular rings with essential socles. These rings were first investigated by Goldie [5] who studied the Artinian case and showed that an indecomposable nonsingular generalized uniserial ring is isomorphic to a full blocked triangular matrix ring over a sfield. The structure of nonsingular rings in which every ideal generated by a primitive idempotent is uniform was determined for the Artinian case by Gordon [6] and Colby and Rutter [2], and for the semiprimary case by Zaks [12]. Nonsingular rings with essential socles and finite identities were characterized by Gordon [7] and the author [10]. All these results were obtained by representing the rings in question as matrix rings. In this paper a matrix representation of arbitrary nonsingular rings with essential socles is found (section 2). The above results are special cases of this representation. A general method for representing rings as matrices is developed in section 1.

Remarks on Centers of Rings

2021 ◽  
Vol 28 (01) ◽  
pp. 1-12
Author(s):  
Juan Huang ◽  
Hailan Jin ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Zhelin Piao
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Full Matrix ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

It is proved that for matrices [Formula: see text], [Formula: see text] in the [Formula: see text] by [Formula: see text] upper triangular matrix ring [Formula: see text] over a domain [Formula: see text], if [Formula: see text] is nonzero and central in [Formula: see text] then [Formula: see text]. The [Formula: see text] by [Formula: see text] full matrix rings over right Noetherian domains are also shown to have this property. In this article we treat a ring property that is a generalization of this result, and a ring with such a property is said to be weakly reversible-over-center. The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains. The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally. We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.

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