Artinian Ring Research Articles

A Generalized Pumping Lemma for Weighted Recognizable Languages

Pumping lemmata are the main tool to prove that a certain language does not belong to a class of languages like the recognizable languages or the context-free languages. Essentially two pumping lemmata exist for the recognizable weighted languages: the classical one for the Boolean semiring (i.e., the unweighted case) and a similar one for fields. A joint generalization of these two pumping lemmata is provided that applies to all (right) Artinian semirings. These are (potentially non-commutative) semirings over which all finitely generated (right) semimodules have a finite bound on the length of chains of strictly increasing (right) subsemimodules. Since Artinian rings are exactly those that satisfy the Descending Chain Condition, the Artinian semirings include all skew fields and naturally also all finite semirings. The new pumping lemma thus covers most previously known pumping lemmata for recognizable weighted languages and a non-exhaustive list of semirings to which the new pumping lemma can be applied is provided as well.

Wide coreflective subcategories and torsion pairs

We revisit a construction of wide subcategories going back to work of Ingalls and Thomas. To a torsion pair in the category R−mod of finitely presented modules over a left artinian ring R, we assign two wide subcategories in the category R−Mod of all R-modules and describe them explicitly in terms of an associated cosilting module. It turns out that these subcategories are coreflective, and we address the question of which wide coreflective subcategories can be obtained in this way. Over a tame hereditary algebra, they are precisely the categories which are perpendicular to collections of pure-injective modules.

On Artinian Rings whose Ideal Graph is a Star

On Artinian Rings whose Ideal Graph is a Star

On invariant submonoids

In the first part of the paper, we provide a fairly complete description of a 2-torsion free simple ring R with involution f such that the skew traces relative to f of all elements of R form a commutative subset. In the remaining part, the results carried out earlier are put to work to delineate the structure of a simple Artinian ring R with involution f, which is equipped with a given non central submonoid M (= multiplicatively closed subset containing 1) preserved under f and under all inner automorphisms of R, and M is such that the skew traces of all elements of M form a commutative subset.

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.

Unifying some classical results on Artinian rings and modules

In this note, we introduce a very crude but natural notion of measure on the class of left R-modules over a ring R. We use this notion to give short proofs of some classical theorems on (left) Artinian rings and modules, due to Akizuki, Anderson, Hopkins, and Levitzki, as well as of some new results.

Some results on the total zero-divisor graph of a commutative ring

PurposeThe purpose of this paper is to characterize a commutative ring R with identity which is not an integral domain such that ZT(R), the total zero-divisor graph of R is connected and to determine the diameter and radius of ZT(R) whenever ZT(R) is connected. Also, the purpose is to generalize some of the known results proved by Duric et al. on the total zero-divisor graph of R.Design/methodology/approachWe use the methods from commutative ring theory on primary decomposition and strong primary decomposition of ideals in commutative rings. The structure of ideals, the primary ideals, the prime ideals, the set of zero-divisors of the finite direct product of commutative rings is used in this paper. The notion of maximal Nagata prime of the zero-ideal of a commutative ring is also used in our discussion.FindingsFor a commutative ring R with identity, ZT(R) is the intersection of the zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R. The zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R are well studied. Hence, we determine necessary and sufficient condition so that ZT(R) agrees with the zero-divisor graph of R (respectively, agrees with the total graph induced by the set of non-zero zero-divisors of R). If Z(R) is an ideal of R, then it is noted that ZT(R) agrees with the zero-divisor graph of R. Hence, we focus on rings R such that Z(R) is not an ideal of R. We are able to characterize R such that ZT(R) is connected under the assumptions that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. With the above assumptions, we are able to determine the domination number of ZT(R).Research limitations/implicationsDuric et al. characterized Artinian rings R such that ZT(R) is connected. In this paper, we extend their result to rings R such that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. As an Artinian ring is isomorphic to the direct product of a finite number of Artinian local rings, we try to characterize R such that ZT(R) is connected under the assumption that R is ta finite direct product of rings R1, R2, … Rn with Z(Ri) is an ideal of Ri for each i between 1 to n. Their result on domination number of ZT(R) is also generalized in this paper. We provide several examples to illustrate our results proved.Practical implicationsThe implication of this paper is that the existing result of Duric et al. is applicable to large class of commutative rings thereby yielding more examples. Moreover, the results proved in this paper make us to understand the structure of commutative rings better. It also helps us to learn the interplay between the ring-theoretic properties and the graph-theoretic properties of the graph associated with it.Originality/valueThe results proved in this paper are original and they provide more insight into the structure of total zero-divisor graph of a commutative ring. This paper provides several examples. Not much work done in the area of total zero-divisor graph of a commutative ring. This paper is a contribution to the area of graphs and rings and may inspire other researchers to study the total zero-divisor graph in further detail.

One Resolution of the Conjecture between the Projective Dimension of a Simple Module S of an Artinian Ring on Zero Radical’s Cube and the First Bifunctor Extension on S

Through concepts from non-commutative algebra and homology, this paper resolves the conjecture that lets the first bifunctor extension to be zero when the projective dimension is finite, for a simple module S of an Artinian ring whose cube of its Jacobson radical is zero and under the condition that any simple module over this ring of finite projective dimension has aradical square zero of the cover projective of its first syzygy. For that, we use a property of the simple module which realizes the minimum of the finite projective dimensions of simple modules. Our main result is presented in the form of a corollary in the case of an Artinian ring with radical cubed zero such that the projective cover of its radical is of Loewy length two and its supremumbeing finite. In the part of discussions, we succeed to show the no loop conjecture through two examples. The first one is about its weak version by taking a quiver algebra A verifying J3 = 0 and without considering that rad2 (P(Ω(S))) = 0 for every simple module, while the second one shows that if the extension quiver has a loop in a simple module then its projective dimension is infinite for every nilpotence index of the Jacobson radical. More importantly, we finally provide a practical third example for our special case.

On the splitting property of rings with restricted class of injectivity domains

There has been a growing research interest in exploring rings whose modules are either injective or far from being injective to a specified degree. As is commonly observed in each of such scenarios, rings can be broken as a direct sum of a semisimple Artinian ring and an indecomposable ring having zero or (essential) homogeneous right socle. Although this phenomenon has become increasingly expected and less surprising with every new study, proving it has never become easier, and in most cases, it has been obtained as a result of considerable efforts. This work aims to bring such studies to a common denominator and tries to explain why this phenomenon occurs each time we deal with the same type of problem.

On semiperfect a -rings

UDC 512.5 A ring is called a right a -ring if every right ideal is automorphism invariant. We describe some properties of a -rings over semiperfect rings. It is shown that an I-finite right a -ring is a direct sum of a semisimple Artinian ring and a basic ring. It is also demonstrated that if R is an indecomposable (as a ring) I-finite right a -ring not simple with nontrivial idempotents such that every minimal right ideal is a right annihilator and S o c ( R R ) = S o c ( R R ) is essential in R R , then R is a quasi-Frobenius ring and it is also a right q -ring.

On the power graphs of semigroups of homogeneous elements of graded semisimple Artinian rings

Let S be a groupoid (magma) with zero 0, and let R = ⊕ s ∈ S R s be a contracted S-graded ring, that is, an S-graded ring with R 0 = 0. By G ( H R ) we denote the undirected power graph of a multiplicative subsemigroup H R = ∪ s ∈ S R s of R, and by G * ( H R ) a graph obtained from G ( H R ) by removing 0 and its incident edges. If Re is a nonzero ring component of R, then G * ( R e ) denotes a subgraph of G * ( H R ) , induced by R e * . In this paper we address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power Graphs: A Survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)]. Namely, let S be torsion-free, that is, s n = t n implies s = t for all s, t ∈ S , and all positive integers n, and let S be 0-cancellative, that is, for all s, t, u ∈ S , su = tu ≠ 0 implies s = t , and us = ut ≠ 0 implies s = t . Also, let R be semisimple Artinian. We prove that if G * ( R e ) is connected for every nonzero ring component Re of R, then the connected components of G * ( H R ) are precisely the graphs G * ( R e ) .

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.

Metric and Upper Dimension of Extended Annihilating-Ideal Graphs

The metric dimension problem is called navigation problem due to its application to robot navigation in space. Further this concept has wide applications in motion planning, sonar and loran station, and so on. In this paper, we study certain results on the metric dimension, upper dimension and resolving number of extended annihilating-ideal graph [Formula: see text] associated to a commutative ring [Formula: see text], denoted by [Formula: see text], [Formula: see text] and [Formula: see text], respectively. Here we prove the finiteness conditions of [Formula: see text] and [Formula: see text]. In addition, we characterize [Formula: see text], [Formula: see text] and [Formula: see text] for artinian rings and the direct product of rings.

On essentially Π-injective modules and rings

In this paper, we study modules having the property that are invariant under some idempotent endomorphisms of its injective envelope. Such modules are called essentially π-injective. It is shown that (1) M is essentially π-injective iff for any essentially finite direct summand X 1 of M and any submodule X 2 of M with X 1 ∩ X 2 = 0 , there exists a direct summand X 0 of M containing X 2 such that M = X 1 ⊕ X 0 , (2) M is essentially π-injective iff M is an ef-extending right R-module and for any decomposition M = M 1 ⊕ M 2 with M 1 essentially finite, M 1 and M 2 are relatively injective, (3) if M is essentially π-injective and R satisfies ACC on right ideals of the form r(m), m ∈ M , then M is a direct sum of uniform submodules. We also describe rings via essentially π-injective modules. It is shown that R is a semisimple artinian ring iff the direct sum of any two essentially π-injective right R-modules is essentially π-injective.

Chern character, infinitesimal Abel-Jacobi map and semi-regularity map

Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology. This enables us to realize (after slight modification) the infinitesimal Abel-Jacobi map as a morphism between tangent spaces of two functors of Artin rings and also enables us to reconstruct the semi-regularity map together with giving a different proof of a theorem of Bloch stating that the semi-regularity map annihilates certain obstructions to embedded deformations of a closed subvariety which is a locally complete intersection.

Inverse polynomials of numerical semigroup rings

We study the defining ideal of a numerical semigroup ring k[H] using the inverse polynomial attached to the Artinian ring k[H]/(th). We give a criterion for H to be symmetric or almost symmetric using the annihilator of the inverse system. Also we give characterizations of symmetric numerical semigroups with small multiplicity and give a new proof of Bresinsky's Theorem for symmetric semigroups generated by 4 elements.

A method for constructing minimal projective resolutions over idempotent subrings

We show how to obtain minimal projective resolutions of finitely generated modules over an idempotent subring Γ e ≔ ( 1 − e ) R ( 1 − e ) \Gamma _e ≔(1-e)R(1-e) of a semiperfect noetherian basic ring R R by a construction inside m o d R \mathsf {mod}\,R . This is then applied to investigate homological properties of idempotent subrings Γ e \Gamma _e under the assumption of R / ⟨ 1 − e ⟩ R/\langle 1-e\rangle being a right artinian ring. In particular, we prove the conjecture by Ingalls and Paquette that a simple module S e ≔ e R / rad ⁡ e R S_e ≔eR /\operatorname {rad}eR with Ext R 1 ⁡ ( S e , S e ) = 0 \operatorname {Ext}_R^1(S_e,S_e) = 0 is self-orthogonal, that is Ext R k ⁡ ( S e , S e ) \operatorname {Ext}^k_R(S_e,S_e) vanishes for all k ≥ 1 k \geq 1 , whenever gldim ⁡ R \operatorname {gldim}R and pdim ⁡ e R ( 1 − e ) Γ e \operatorname {pdim}eR(1-e)_{\Gamma _e} are finite. Indeed, a slightly more general result is established, which applies to sandwiched idempotent subrings: Suppose e ∈ R e \in R is an idempotent such that all idempotent subrings Γ \Gamma sandwiched between Γ e \Gamma _e and R R , that is Γ e ⊆ Γ ⊆ R \Gamma _e \subseteq \Gamma \subseteq R , have finite global dimension. Then the simple summands of S e S_e can be numbered S 1 , … , S n S_1, \dots , S_n such that Ext R k ⁡ ( S i , S j ) = 0 \operatorname {Ext}_R^k(S_i, S_j) = 0 for 1 ≤ j ≤ i ≤ n 1 \leq j \leq i \leq n and all k > 0 k > 0 .

Certain invariant multiplicative subset of a simple Artinian ring with involution. II

Certain invariant multiplicative subset of a simple Artinian ring with involution. II

The Finitistic Dimension and Global Dimension of an Artinian Ring with a Dualizing Complex

The Finitistic Dimension and Global Dimension of an Artinian Ring with a Dualizing Complex

Co-retractable Modules and Semi-Simplicity Property

Co-retractable modules appeared in 2006 by Clark, Lomp and Vanaja. In this article, we present and study the relationships between two known concepts namely semi-simple modules and Co-retractable module. Our concern in this article is to develop some properties of Co-retractable module and related to other classes of modules such as multiplication module, indecomposable module and projective module. We prove that if M is a multiplication R-module with annR(M) is a prime ideal and the Socal of M is a direct summand of M, this means M is a retractable module. Also, if M is an indecomposable multiplication R-module with annR(M) is a prime ideal of Noetherian Ring R and every direct sum of modules with summand intersection property has summand sum property, then M is a co-retractable module. Finally, if M is a simple quasi-Dedekind module over the Artinian Ring R with rad(R) = 0 and annR(M) is a prime ideal of R, then M is a co-retractable module.