On the girth of regular digraph of ideals of an Artinian ring

AbstractA number of constructions are given for arc-transitive digraphs, based on modifications of permutation representations of finite groups. In particular, it is shown that for every positive integer s and for any transitive permutation group p of degree k, there are infinitely many examples of a finite k-regular digraph with a group of automorphisms acting transitively on s-arcs (but not on (s + 1)-arcs), such that the stabilizer of a vertex induces the action of P on the out-neighbour set.

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The Weakly Nilpotent Graph of a Commutative Ring

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-096-1 ◽

2017 ◽

Vol 60 (2) ◽

pp. 319-328

Author(s):

Soheila Khojasteh ◽

Mohammad Javad Nikmehr

Keyword(s):

Commutative Ring ◽

Chromatic Number ◽

Disjoint Union ◽

Independence Number ◽

Artinian Ring ◽

Clique Number ◽

Commutative Rings ◽

Unicyclic Graph ◽

Nilpotent Elements ◽

Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

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Homological Aspects of Semigroup Gradings on Rings and Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1999-022-5 ◽

1999 ◽

Vol 51 (3) ◽

pp. 488-505 ◽

Cited By ~ 3

Author(s):

W. D. Burgess ◽

Manuel Saorín

Keyword(s):

Euler Characteristic ◽

Division Ring ◽

Artinian Ring ◽

Global Dimension ◽

Monomial Algebra ◽

Finite Dimensional ◽

Cartan Matrices ◽

Torsion Free ◽

Rings And Algebras ◽

Finite Dimensional Algebras

AbstractThis article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup Σ. Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the Σ-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties.A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert Σ-series in the associated path incidence ring.The rationality of the Σ-Euler characteristic, the Hilbert Σ-series and the Poincaré-Betti Σ-series is studied when Σ is torsion-free commutative and A is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.

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A GENERALIZATION OF BAER'S LOWER NILRADICAL FOR MODULES

Journal of Algebra and Its Applications ◽

10.1142/s0219498807002284 ◽

2007 ◽

Vol 06 (02) ◽

pp. 337-353 ◽

Cited By ~ 10

Author(s):

MAHMOOD BEHBOODI

Keyword(s):

Prime Ring ◽

Artinian Ring ◽

Prime Radical ◽

Nilpotent Elements ◽

Noetherian Domain ◽

Prime Submodules ◽

Strongly Nilpotent ◽

Commutative Domain ◽

Torsion Modules

Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals [Formula: see text] and for all submodules N ⊆ M, [Formula: see text] implies that [Formula: see text] or [Formula: see text]. We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.rad R(M)] and Baer's lower nilradical for a module [denoted by Nil *(RM)]. For a module RM, cl.rad R(M) is defined to be the intersection of all classical prime submodules of M and Nil *(RM) is defined to be the set of all strongly nilpotent elements of M (defined later). It is shown that, for any projective R-module M, cl.rad R(M) = Nil *(RM) and, for any module M over a left Artinian ring R, cl.rad R(M) = Nil *(RM) = Rad (M) = Jac (R)M. In particular, if R is a commutative Noetherian domain with dim (R) ≤ 1, then for any module M, we have cl.rad R(M) = Nil *(RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general modules reduces to that of torsion modules. Moreover, over an FBN prime ring R with dim (R) ≤ 1 (or over a commutative domain R with dim (R) ≤ 1), every semiprime submodule of any module is an intersection of classical prime submodules.

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The Classification of the Annihilating-Ideal Graphs of Commutative Rings

Algebra Colloquium ◽

10.1142/s1005386714000200 ◽

2014 ◽

Vol 21 (02) ◽

pp. 249-256 ◽

Cited By ~ 20

Author(s):

G. Aalipour ◽

S. Akbari ◽

M. Behboodi ◽

R. Nikandish ◽

M. J. Nikmehr ◽

...

Keyword(s):

Commutative Ring ◽

Chromatic Number ◽

Artinian Ring ◽

Clique Number ◽

Commutative Rings ◽

Vertex Set

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

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Example of an Injective Module which is Not Nice

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-028-4 ◽

1974 ◽

Vol 17 (1) ◽

pp. 133-134

Author(s):

Gerhard O. Michler

Keyword(s):

Artinian Ring ◽

Injective Module ◽

Finite Topology ◽

Factor Module ◽

Ring Of Quotients

In [1] Lambek calls the injective R-module I nice if every torsionfree factor module of the ring of quotients Q of R with respect to lis divisible. If lis nice then g is a dense subring of the bicommutator BicRI of I with respect to the finite topology (see [1, Proposition 2]). We now give an example of an injective R-module over an Artinian ring R which is not nice. Since R is Artinian, Q=BicRI, by Proposition B of [1].Before we give the example, we state the following, which depends on [2] for terminology.

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The graded Cartan matrix and global dimension of 0-relations algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500026742 ◽

1987 ◽

Vol 30 (3) ◽

pp. 351-362 ◽

Cited By ~ 5

Author(s):

W. D. Burgess

Keyword(s):

Artinian Ring ◽

Global Dimension ◽

Cartan Matrix ◽

Integral Matrix ◽

Finite Global Dimension ◽

Composition Factors

The Cartan matrix C of a left artinian ring A, with indecomposable projectives P1,…,Pn and corresponding simples Si=Pi/JPi, is an n×n integral matrix with entries Cij, the number of copies of the simple sj which appear as composition factors of Pi. A relationship between the invertibility of this matrix (as an integral matrix) and the finiteness of the global dimension has long been known: gl dim A < ∞⇒det C = ± 1 (Eilenberg [3]). More recently Zacharia [9] has shown that gl dim A ≦ 2⇒det C = 1, and in fact no rings of finite global dimension are known with det C = −1. The converse, det C = l⇒gl dim A < ∞, is false, as easy examples show ([[1) or [3]). However if A is left serial, gl dim A < ∞iff det C = l [1]. If A = ⊕n ≧ 0 An is ℤ-graded and the radical J = ⊕n ≧ 0 An, Wilson [8] calls such rings positively graded. Here there is a graded Cartan matrix with entries from ℤ[X] and gl dim A < ∞⇒det = 1 and, hence, det C = l [8, Prop. 2.2].

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Rings whose Indecomposable Injective Modules are Uniserial

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-055-3 ◽

1982 ◽

Vol 34 (4) ◽

pp. 797-805 ◽

Cited By ~ 7

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.

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