Commutative Artinian Ring Research Articles

On rings whose prime ideal sum graphs are line graphs

Let [Formula: see text] be a commutative ring with unity. The prime ideal sum graph of the ring [Formula: see text] is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph.

Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs

Let R be a commutative ring with unity. The prime ideal sum graph PIS ( R ) of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent if and only if I + J is a prime ideal of R. In this paper, we study some interplay between algebraic properties of rings and graph-theoretic properties of their prime ideal sum graphs. In this connection, we classify non-local commutative Artinian rings R such that PIS ( R ) is of crosscap at most two. We prove that there does not exist a non-local commutative Artinian ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative Artinian rings of genus one prime ideal sum graphs.

On the genus of strong annihilating-ideal graph of commutative rings

Let [Formula: see text] be a commutative ring with unity and [Formula: see text] be the set of annihilating-ideals of [Formula: see text]. The strong annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is an undirected graph with vertex set [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. In this paper, first we characterize commutative Artinian rings whose strong annihilating-ideal graph is isomorphic to some well-known graphs and then we classify commutative Artinian rings whose strong annihilating-ideal graph is planar, toroidal or projective.

Auslander-Reiten n-exangles and morphisms determined by objects in n-exangulated categories

Let k be commutative artinian ring and be an Ext-finite, Krull-Schmidt n-exangulated category. We define two additive subcategories and of in terms of the representation functors from the stable category of to k-modules. Moreover, we introduce two quasi-inverse functors and . Furthermore, we investigate the subcategories and in terms of morphisms determined by objects, and then give equivalent characterizations for the existence of Auslander-Reiten n-exangles.

Higher Auslander-Reiten sequences via morphisms determined by objects

Let be an Ext-finite, Krull-Schmidt and k-linear n-exangulated category with k a commutative artinian ring. In this note, we define two additive subcategories and of in terms of the representable functors from the stable category of to the category of finitely generated k-modules. Moreover, we show that there exists an equivalence between the stable categories of these two full subcategories. Finally, we give some equivalent characterizations on the existence of Auslander-Reiten n-exangles via determined morphisms. These results unify and extend their works by Jiao–Le for exact categories, Zhao–Tan–Huang for extriangulated categories, Xie–Liu–Yang for n-abelian categories.

Auslander–Reiten–Serre duality for n-exangulated categories

Let [Formula: see text] be an [Formula: see text]-finite, Krull–Schmidt and [Formula: see text]-linear [Formula: see text]-exangulated category with [Formula: see text] a commutative artinian ring. In this note, we prove that [Formula: see text] has Auslander–Reiten–Serre duality if and only if [Formula: see text] has Auslander–Reiten [Formula: see text]-exangles. Moreover, we also give an equivalent condition for the existence of Serre duality (which is a special type of Auslander–Reiten–Serre duality). Finally, assume further that [Formula: see text] has Auslander–Reiten–Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection and the Auslander–Reiten–Serre duality.

The existence of n-Auslander–Reiten sequences via determined morphisms

Let [Formula: see text] be a commutative artinian ring and [Formula: see text] a small Ext-finite Krull–Schmidt [Formula: see text]-abelian [Formula: see text]-category with enough projectives and injectives. We introduce two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text] in terms of the representable functors from the stable category of [Formula: see text] to category of finitely generated [Formula: see text]-modules. Moreover, we define two additive functors [Formula: see text] and [Formula: see text], which are mutually quasi-inverse equivalences between the stable categories of this two full subcategories. We give an equivalent characterization on the existence of [Formula: see text]-Auslander–Reiten sequences using determined morphisms.

Cut vertices in comaximal graph of a commutative Artinian ring

Cut vertices in comaximal graph of a commutative Artinian ring

On the problem of zero-divisors in Artinian rings

Let R be a commutative Artinian ring and let be the compressed zero-divisor graph associated to R. The question of when the clique number was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff. They proved that if (where is the largest length of any of its chains of ideals), then When they gave an example of a local ring R where is possible by using the trivial extension of an Artinian local ring by its dualizing module. The question of what happens when was stated as an open question. We show that if then We first reduce the problem to the case of a local ring We then enumerate all possible Hilbert functions of R and show that the k-vector space admits a symmetric bilinear form in some cases of the Hilbert function. This allows us to relate the orthogonality in the bilinear space with the structure of zero-divisors in R. For instance, in the case when is principal and we show that R is Gorenstein if and only if the symmetric bilinear form on is non-degenerate. Moreover, in the case when our techniques also yield a simpler and shorter proof of the finiteness of avoiding, for instance, the Cohen structure theorem.

On Perfect Co-Annihilating-Ideal Graph of a Commutative Artinian Ring

Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by AR, is a graph whose vertex set is the set of all non-zero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann(I) ∩ Ann(J) = (0). In this paper, we characterize all Artinian rings for which both of the graphs AR and AR (the complement of AR), are chordal. Moreover, all Artinian rings whose AR (and thus AR) is perfect are characterized.

A bijection triangle in extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories, and recently Iyama, Nakaoka and Palu investigated Auslander-Reiten theory in terms of Auslander-Reiten-Serre duality in extriangulated categories. In this paper, we introduce the notion of Serre duality, as a special type of Auslander-Reiten-Serre duality, and then give an equivalent condition for the existence of Serre duality. On the other hand, we study a bijection triangle in extriangulated categories which involves the restricted Auslander bijection and Auslander-Reiten-Serre duality. Let R be a commutative artinian ring. We show that the restricted Auslander bijection holds true in Hom-finite R-linear Krull-Schmidt extriangulated categories having Auslander-Reiten-Serre duality, and especially obtain the Auslander bijection in Hom-finite R-linear Krull-Schmidt extriangulated categories having Serre duality. We also give some applications, and in particular, we show that a conjecture given by Ringel holds true in this setting.

Perfect unit graphs of commutative Artinian rings

Let R be a commutative ring with identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R and two distinct vertices x and y are adjacent if and only if $$x+y$$ is a unit of R. In this paper, perfect unit graphs of Artinian rings are investigated.

The coloring of the cozero-divisor graph of a commutative ring

Let [Formula: see text] be a commutative ring with unity. The cozero-divisor graph of [Formula: see text] denoted by [Formula: see text] is a graph with the vertex set [Formula: see text], where [Formula: see text] is the set of all nonzero and non-unit elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote the clique number and the chromatic number of [Formula: see text], respectively. In this paper, we prove that if [Formula: see text] is a finite commutative ring, then [Formula: see text] is perfect. Also, we prove that if [Formula: see text] is a commutative Artinian non-local ring and [Formula: see text] is finite, then [Formula: see text]. For Artinian local ring, we obtain an upper bound for the chromatic number of cozero-divisor graph. Among other results, we prove that if [Formula: see text] is a commutative ring, then [Formula: see text] is a complete bipartite graph if and only if [Formula: see text], where [Formula: see text] and [Formula: see text] are fields. Moreover, we present some results on the complete [Formula: see text]-partite cozero-divisor graphs.

Almost split triangles and morphisms determined by objects in extriangulated categories

Let (C,E,s) be an Ext-finite, Krull-Schmidt and k-linear extriangulated category with k a commutative artinian ring. We define an additive subcategory Cr (respectively, Cl) of C in terms of the representable functors from the stable category of C modulo s-injectives (respectively, s-projectives) to k-modules, which consists of all s-projective (respectively, s-injective) objects and objects isomorphic to direct summands of finite direct sums of all third (respectively, first) terms of almost split s-triangles. We investigate the subcategories Cr and Cl in terms of morphisms determined by objects, and then give equivalent characterizations on the existence of almost split s-triangles.

A note on comaximal ideal graph of commutative rings

Let R be a commutative ring with identity. The comaximal ideal graphG(R) of R is a simple graph with its vertices are the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1+I2=R. In this paper, a dominating set of G(R) is constructed using elements of the center when R is a commutative Artinian ring. Also we prove that the domination number of G(R) is equal to the number of factors in the Artinian decomposition of R. Also, we characterize all commutative Artinian rings(non local rings) with identity for which G(R) is planar.

Duality and contravariant functors in the representation theory of artin algebras

We know that the model theory of modules leads to a way of obtaining definable categories of modules over a ring [Formula: see text] as the kernels of certain functors [Formula: see text] rather than of functors [Formula: see text] which are given by a pp-pair. This paper will give various algebraic characterizations of these functors in the case that [Formula: see text] is an artin algebra. Suppose that [Formula: see text] is an artin algebra. An additive functor [Formula: see text] preserves inverse limits and [Formula: see text] is finitely presented if and only if there is a sequence of natural transformations [Formula: see text] for some [Formula: see text] which is exact when evaluated at any left [Formula: see text]-module. Any additive functor [Formula: see text] with one of these equivalent properties has a definable kernel, and every definable subcategory of [Formula: see text] can be obtained as the kernel of a family of such functors. In the final section, a generalized setting is introduced, so that our results apply to more categories than those of the form [Formula: see text] for an artin algebra [Formula: see text]. That is, our results are extended to those locally finitely presented [Formula: see text]-linear categories whose finitely presented objects form a dualizing variety, where [Formula: see text] is a commutative artinian ring.

Comparison of graphs associated to a commutative Artinian ring

Let R be a commutative ring with $$1\ne 0$$ and the additive group $$R^+$$ . Several graphs on R have been introduced by many authors, among zero-divisor graph $$\Gamma _1(R)$$ , co-maximal graph $$\Gamma _2(R)$$ , annihilator graph AG(R), total graph $$ T(\Gamma (R))$$ , cozero-divisors graph $$\Gamma _\mathrm{c}(R)$$ , equivalence classes graph $$\Gamma _\mathrm{E}(R)$$ and the Cayley graph $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when R is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if R has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results, we prove that for a commutative finite ring R with $$|\mathrm{Max}(R)|=n \ge 3$$ , $$ \Gamma _1(R) \simeq \Gamma _2(R)$$ if and only if $$R\simeq \mathbb {Z}^n_2$$ ; if and only if $$\Gamma _1(R) \simeq \Gamma _\mathrm{E}(R)$$ . Also the annihilator graph is identical to the cozero-divisor graph if and only if R is a Frobenius ring.

On the nil-graph of ideals of commutative Artinian rings

Let [Formula: see text] be a commutative ring with identity and Nil[Formula: see text] be the ideal consisting of all nilpotent elements of [Formula: see text]. Let [Formula: see text] [Formula: see text] [Formula: see text]. The nil-graph of ideals of [Formula: see text] is defined as the graph [Formula: see text] whose vertex set is the set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we discuss some properties of nil-graph of ideals concerning connectedness, split and claw free. Also we characterize all commutative Artinian rings [Formula: see text] for which the nil-graph [Formula: see text] has genus 2.

Cayley sum graph of ideals of commutative rings

Let [Formula: see text] be a commutative ring, [Formula: see text] the set of all ideals of [Formula: see text] and [Formula: see text], a subset of [Formula: see text]. The Cayley sum graph of ideals of [Formula: see text], denoted by Cay[Formula: see text], is a simple undirected graph with vertex set is the set [Formula: see text] and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text], for some [Formula: see text] in [Formula: see text]. In this paper, we study connectedness, Eulerian and Hamiltonian properties of Cay[Formula: see text]. Moreover, we characterize all commutative Artinian rings [Formula: see text] whose Cay[Formula: see text] is toroidal.

The classi cation of rings with its genus of class of graphs

Let $R$ be a commutative ring, $I$ be a proper ideal of $R$, and $S(I)=\{a\in R : ra\in I \text{ for some } r\in R\sm I\}$ be the set of all elements of $R$ that are not prime to $I$. The total graph of $R$ with respect to $I$, denoted by $T(\Gamma_I(R))$, is the simple graph with all elements of $R$ as vertices, and for distinct $x,y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x+y\in S(I)$. In this paper, we determine all isomorphic classes of commutative Artinian rings whose ideal-based total graph has genus at most two.