scholarly journals Artinian bimodule with quasi-Frobenius bimodule of translations

2019 ◽  
Vol 29 (2) ◽  
pp. 103-119
Author(s):  
Aleksandr A. Nechaev ◽  
Vadim N. Tsypyschev
Keyword(s):  
Commutative Ring ◽  
Recurrent Sequence ◽  
Artinian Rings ◽  
Linear Recurrent Sequence ◽  
Left And Right ◽  
Additional Assumptions

Abstract The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule AMB over left- and right-Artinian rings A and B, respectively, is associated with the equivalent bimodule of translations CMZ, where C is the multiplicative ring of the bimodule AMB and Z is its center, and the relation between the quasi-Frobenius conditions for the bimodules AMB and CMZ is studied. It is demonstrated that, in the general case, the fact that AMB is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule CMZ. However, under some additional assumptions it can be shown that if CMZ is a quasi-Frobenius bimodule, then the bimodule AMB is quasi-Frobenius as well.

On commutative rings with uniserial dimension

2014 ◽  
Vol 14 (01) ◽  
pp. 1550008 ◽  
Author(s):  
A. Ghorbani ◽  
Z. Nazemian
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Artinian Rings ◽  
Uniform Dimension ◽  
Uniserial Modules

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

Rings whose Indecomposable Injective Modules are Uniserial

1982 ◽  
Vol 34 (4) ◽  
pp. 797-805 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Artinian Ring ◽  
Dimensional Algebra ◽  
Artinian Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Finite Dimensional ◽  
Uniserial Modules ◽  
Left And Right ◽  
Finitely Generated Modules ◽  
Algebra Over A Field

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

Hereditary artinian rings of finite representation type and extensions of simple artinian rings

1987 ◽  
Vol 102 (3) ◽  
pp. 411-420 ◽  
Author(s):  
Aidan Schofield
Keyword(s):  
Artinian Ring ◽  
Representation Type ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Artinian Rings ◽  
Finite Dimensional ◽  
Coxeter Diagram ◽  
Skew Fields ◽  
Left And Right ◽  
Natural Way

In [1], Dowbor, Ringel and Simson consider hereditary artinian rings of finite representation type; it is known that if A is an hereditary artinian algebra of finite representation type, finite-dimensional over a field, then it corresponds to a Dynkin diagram in a natural way; they show that an hereditary artinian ring of finite representation type corresponds to a Coxeter diagram. However, in order to construct an hereditary artinian ring of finite representation type corresponding to a Coxeter diagram that is not Dynkin, they show that it is necessary though not sufficient to find an extension of skew fields such that the left and right dimensions are both finite but are different. No examples of such skew fields were known at the time. In [3], I constructed such extensions, and the main aim of this paper is to extend the methods of that paper to construct an extension of skew fields having all the properties needed to construct an hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5).

Classification of rings with toroidal co-annihilating-ideal graphs

2020 ◽  
pp. 2150006
Author(s):  
Rana Khoeilar ◽  
Jafar Amjadi
Keyword(s):  
Commutative Ring ◽  
Artinian Rings ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity. The co-annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text]. In this paper, we study the planarity and genus of [Formula: see text]. In particular, we characterize all Artinian rings [Formula: see text] for which the genus of [Formula: see text] is zero or one.

C-RINGS, COPRODUCTS, AND REFLECTION FUNCTORS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350071
Author(s):  
EDWARD L. GREEN ◽  
NICHOLAS A. LOEHR ◽  
ALLEN N. PELLEY
Keyword(s):  
Commutative Ring ◽  
Full Subcategory ◽  
Path Algebra ◽  
Ring Homomorphisms ◽  
Left And Right ◽  
Category Of Modules ◽  
Reflection Functors

The paper begins with a detailed study of the category of modules over two different rings using a coproduct construction. If C is a commutative ring and R and S are rings together with ring homomorphisms from C to R and C to S, then we show that the category of C-modules that are also left R-modules and right S-modules is equivalent to the category of left modules over the coproduct of R and S op in an appropriate category. Letting K denote a field, we apply this to show that the category of K-representations of a quiver that is reflection equivalent to a path algebra KΓ is equivalent to a full subcategory of left and right K-representations of KΓ.

scholarly journals Non-local rings whose ideals are all quasi-injective

1972 ◽  
Vol 6 (1) ◽  
pp. 45-52 ◽  
Author(s):  
G. Ivanov
Keyword(s):  
Finite Number ◽  
Local Rings ◽  
Artinian Rings ◽  
Direct Sum ◽  
Non Local ◽  
Left And Right

A ring is a left Q-ring if all of its left ideals are quasi-injective. For an integer m ≤ 2, a sfield D, and a null D-algebra V whose left and right D-dimensions are both equal to one, let H(m, D, V) be the ring of all m x m matrices whose only non-zero entries are arbitrary elements of D along the diagonal and arbitrary elements of V at the places (2, 1), …, (m, m-l) and (l, m). We show that the only indecomposable non-local left Q-rings are the simple artinian rings and the rings H(m, D, V). An arbitrary left Q-ring is the direct sum of a finite number of indecomposable non-local left Q-rings and a Q-ring whose idempotents are all central.

The spectrum subgraph of the annihilating-ideal graph of a commutative ring

2015 ◽  
Vol 14 (08) ◽  
pp. 1550130
Author(s):  
R. Taheri ◽  
M. Behboodi ◽  
A. Tehranian
Keyword(s):  
Commutative Ring ◽  
Complete Graph ◽  
Connected Graph ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Star Graph ◽  
Induced Subgraph ◽  
Artinian Rings ◽  
Spectrum Graph

In this paper we introduce and study the spectrum graph of a commutative ring R, denoted by 𝔸𝔾s(R), that is, the graph whose vertices are all non-zero prime ideals of R with non-zero annihilator and two distinct vertices P1, P2 are adjacent if and only if P1P2 = (0). This is an induced subgraph of the annihilating-ideal graph 𝔸𝔾(R) of R. Among other results, we present the structures of all graphs which can be realized as the spectrum graph of a commutative ring. Then we show that for a non-domain Noetherian ring R, 𝔸𝔾s(R), is a connected graph if and only if 𝔸𝔾s(R) is a star graph if and only if 𝔸𝔾s(R) ≅ K1, K2 or K1,∞, where Kn is a complete graph with n vertices and K1,∞ is a star graph with infinite vertices. Also, we completely characterize the spectrum graphs of Artinian rings. Finally, as an application, we present some relationships between the annihilating-ideal graph and its spectrum subgraph.

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