Left Artinian Ring Research Articles

Graded (quasi-)Frobenius rings

In order to study graded Frobenius algebras from a ring theoretical perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings and a shift-version of the latter ones, and we investigate the structure and representations of such objects. We need to revisit graded simple graded left Artinian rings, graded semisimple rings, and to provide graded versions of certain results concerning the Jacobson radical, the singular radical, and their connection to finiteness conditions and injectivity. We prove a structure result for (shift-)graded Frobenius rings.

Skew McCoy rings and σ-compatibility

In this paper, we study [Formula: see text]-skew McCoy rings under the [Formula: see text]-compatible or the [Formula: see text]-semicompatible conditions. We show that if [Formula: see text] is a semicommutative right or left artinian ring which is [Formula: see text]-semicompatible with an epimorphism [Formula: see text], then the Jacobson radical [Formula: see text] is [Formula: see text]-skew McCoy. As a corollary, we get that the Jacobson radical of a semicommutative artinian ring is right McCoy. We also show that every [Formula: see text]-compatible right duo ring is [Formula: see text]-skew McCoy and that for [Formula: see text]-compatible regular rings, the notions of the [Formula: see text]-skew McCoy and the right McCoy coincide. In addition, we show that every [Formula: see text]-semicompatible semicommutative ring is linearly [Formula: see text]-skew Camillo and that every matrix ring over a division ring is linearly [Formula: see text]-skew Camillo for any endomorphism [Formula: see text].

On weak co-$H$-sequences in two-sided Harada rings

In [7] (and [8]) M. Harada studied a left artinian ring $R$ such that every non-small left $R$-module contains a non-zero injective submodule. And in [12] K. Oshiro called the ring a left Harada ring (abbreviated left $H$-ring) and further studied the ring in [13], [14], [15], $\ldots$. We can see many results on left Harada rings in [4] and many equivalent conditions in [3, Theorem B]. In [5] we give new concepts “co-$H$-sequence” and “$H$-epimorphism” to study two-sided Harada rings. In this paper, we intruduce another new concept “weak co-$H$-sequence”. And we further study two-sided Harada rings.

On infinite MacWilliams rings and minimal injectivity conditions

We provide a complete answer to the problem of characterizing left Artinian rings which satisfy the (left or right) MacWilliams extension theorem for linear codes, generalizing results of Iovanov [J. Pure Appl. Algebra 220 (2016), pp. 560–576] and Schneider and Zumbrägel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961] and answering the question of Schneider and Zumbragel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961]. We show that they are quasi-Frobenius rings, and are precisely the rings which are a product of a finite Frobenius ring and an infinite quasi-Frobenius ring with no non-trivial finite modules (quotients). For this, we give a more general “minimal test for injectivity” for a left Artinian ring A A : we show that if every injective morphism Σ k → A \Sigma _k\rightarrow A from the k k ’th socle of A A extends to a morphism A → A A\rightarrow A , then the ring is quasi-Frobenius. We also give a general result under which if injective maps N → M N\rightarrow M from submodules N N of a module M M extend to endomorphisms of M M (pseudo-injectivity), then all such morphisms N → M N\rightarrow M extend (quasi-injectivity) and obtain further applications.

Finitistic dimension conjectures via Gorenstein projective dimension

It is a well-known result of Auslander and Reiten that contravariant finiteness of the class P∞fin (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, we examine in this work the Gorenstein counterpart of Auslander–Reiten condition, namely contravariant finiteness of the class GP∞fin (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class GP∞fin implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander–Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in the sense that contravariant finiteness of the class GP∞fin implies contravariant finiteness of the class P∞fin over any Artin algebra, and the converse holds for Artin algebras over which the class GP0fin (of finitely generated Gorenstein projective modules) is contravariantly finite.

Line graph associated to the intersection graph of ideals of rings

Let R be a ring with unity and I(R) ∗ are all non-trivial left ideals of R. The intersection graph of ideals of R is denoted by G(R) is an undirected simple graph with vertex set I(R) ∗ and two distinct vertices I and J are adjacent if and only if I∩J ≠ 0. In this article, we investigate some basic properties of the line graph associated to G(R), denoted by L(G(R)). Moreover, we investigate completeness, unicyclicness, bipartiteness, planarity, outerplanarity, ring graph, diameter, girth and clique of L(G(Z n )). We also investigate some basic properties of L(G(R)) for left Artinian ring and finally, we determine the domination number and bondage number of L(G(Z n )).

Line graph associated to the intersection graph of ideals of rings

Let R be a ring with unity and I(R)∗ are all non-trivial left ideals of R. The intersection graph of ideals of R is denoted by G(R) is an undirected simple graph with vertex set I(R)∗ and two distinct vertices I and J are adjacent if and only if I∩J ≠ 0. In this article, we investigate some basic properties of the line graph associated to G(R), denoted by L(G(R)). Moreover, we investigate completeness, unicyclicness, bipartiteness, planarity, outerplanarity, ring graph, diameter, girth and clique of L(G(Zn)). We also investigate some basic properties of L(G(R)) for left Artinian ring and finally, we determine the domination number and bondage number of L(G(Zn)).

MacWilliams’ extension theorem for infinite rings

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient if and only if it admits a finitarily left torsion-free character, if and only if the Pontryagin dual of the regular left module is almost monothetic. In conclusion, an Artinian ring has the MacWilliams property if and only if it is finitarily Frobenius, i.e., it is quasi-Frobenius and its finitary socle embeds into the semisimple quotient.

Noetherian Rings whose Modules are Prime Serial

A theorem due to Nakayama and Skornyakov states that “a ring R is an Artinian serial ring if and only if all left R-modules are serial” and a theorem due to Warfield state that “a Noetherian ring R is serial if and only if every finitely generated left R-module is serial”. We say that an R-module M is prime uniserial (℘-uniserial, for short) if for every pair P, Q of prime submodules of M either \(P\subseteq Q\) or \(Q\subseteq P\), and we say that M is prime serial (℘-serial, for short) if it is a direct sum of ℘-uniserial modules. Therefore, two interesting natural questions of this sort are: “Which rings have the property that every module is ℘-serial?” and “Which rings have the property that every finitely generated module is ℘-serial?” Most recently, in our paper, Prime uniserial modules and rings (submitted), we considered these questions in the context of commutative rings. The goal of this paper is to answer these questions in the case R is a Noetherian ring in which all idempotents are central or R is a left Artinian ring.

Left localizations of left Artinian rings

For an arbitrary left Artinian ring [Formula: see text], explicit descriptions are given of all the left denominator sets [Formula: see text] of [Formula: see text] and left localizations [Formula: see text] of [Formula: see text]. It is proved that, up to [Formula: see text]-isomorphism, there are only finitely many left localizations and each of them is an idempotent localization, i.e. [Formula: see text] and [Formula: see text] where [Formula: see text] is a left denominator set of [Formula: see text] and [Formula: see text] is an idempotent. Moreover, the idempotent [Formula: see text] is unique up to a conjugation. It is proved that the number of maximal left denominator sets of [Formula: see text] is finite and does not exceed the number of isomorphism classes of simple left [Formula: see text]-modules. The set of maximal left denominator sets of [Formula: see text] and the left localization radical of [Formula: see text] are described.

On a ring structure related to annihilators

In this note, we focus our attention on a new ring structure related to annihilators, and consider a ring property that contains many kinds of ring classes, introducing right ZAFS. This property is shown to be not left-right symmetric but left-right symmetric for left or right Artinian rings. The left (right) ZAFS property is shown to pass to Ore extensions with automorphisms. The left (respectively, right) ZAFS property is shown to pass also to classical left (respectively, right) quotient rings, yielding that semiprime right Goldie rings are ZAFS.

Rings whose cyclic modules are pure-injective or pure-projective

A famous theorem of algebra due to Osofsky states that “if every cyclic left R-module is injective, then R is semisimple”. Therefore, a natural question of this sort is: “What is the class of rings R for which every cyclic left R-module is pure-injective or pure-projective?” The goal of this paper is to answer this question. For instance, we show that if every cyclic left R-module is pure-injective, then R is a left perfect ring. As a consequence, a commutative coherent ring R is Artinian if and only if every cyclic R-module is pure-injective. Also, a commutative ring R is pure-semisimple (i.e., every R-module is pure-injective) if and only if all cyclic R-modules and all indecomposable R-modules are pure-injective. We obtain some generalizations of Osofsky's theorem in the cases R is semiprimitive or commutative coherent or a commutative semiprime Goldie ring. Finally, we show that a ring R is left Noetherian if and only if every cyclic left R-module is pure-projective. As a corollary of this result we obtain: if every cyclic left R-module is pure-injective and pure-projective, then R is a left Artinian ring. The converse is also true when R is commutative.

CLASSICAL COMPLETELY PRIME SUBMODULES

We define and characterize classical completely prime submodules which are a generalization of both completely prime ideals in rings and reduced modules (as defined by Lee and Zhou in [18]). A comparison of these submodules with other “prime” submodules in literature is done. If Rad(M) is the Jacobson radical of M and $\beta_cl^c(M)$ the classical completely prime radical of M, we show that for modules over left Artinian rings R, Rad(M) $\subseteq \beta_cl^c(M)$ and Rad($_R$R) = $\beta_cl^c(_RR)$ .

On the Unit Graph of a Non-commutative Ring

Let R be a ring with non-zero identity. The unit graph G(R) of R is a graph with elements of R as its vertices and two distinct vertices a and b are adjacent if and only if a + b is a unit element of R. It was proved that if R is a commutative ring and 𝔪 is a maximal ideal of R such that |R/𝔪| = 2, then G(R) is a complete bipartite graph if and only if (R, 𝔪) is a local ring. In this paper we generalize this result by showing that if R is a ring (not necessarily commutative), then G(R) is a complete r-partite graph if and only if (R, 𝔪) is a local ring and r = |R/𝔪| = 2n for some n ∈ ℕ or R is a finite field. Among other results we show that if R is a left Artinian ring, 2 ∈ U(R) and the clique number of G(R) is finite, then R is a finite ring.

Minimal faithful modules over Artinian rings

Let R be a left Artinian ring, and M a faithful left R-module which is minimal, in the sense that no proper submodule or proper homomorphic image of M is faithful. If R is local, and socle(R) is central in R, we show that length(M/J(R)M)+length(socle(M))$\leq$ length(socle(R))+1, strengthening a result of T. Gulliksen. Most of the rest of the paper is devoted to the situation where the Artinian ring R is not necessarily local, and does not necessarily have central socle. In the case where R is a finite-dimensional algebra over an algebraically closed field, we get an inequality similar to the above, with the length of socle(R) interpreted as its length as a bimodule, and with the final summand +1 replaced by the Euler characteristic of a bipartite graph determined by the structure of socle(R). More generally, that inequality holds if, rather than assuming k algebraically closed, we assume that R/J(R) is a direct product of full matrix algebras over k, and exclude the case where k has small finite cardinality. Examples show that the restriction on the cardinality of k is needed; we do not know whether versions of our result hold with other hypotheses significantly weakened. The situation for faithful modules with only one minimality property, i.e., having no faithful proper submodules or having no faithful proper homomorphic images, is more straightforward: The length of M/J(R)M in the former case, and of socle(M) in the latter, is $\leq$ length(socle(R)) (again meaning length as a bimodule). The final section, essentially independent of the rest of the note, obtains these bounds, and shows that every faithful module over a left Artinian ring has a faithful submodule with the former minimality condition and a faithful factor module with the latter. The proofs involve some nice general results on decompositions of modules.

The largest strong left quotient ring of a ring

For an arbitrary ring R, the largest strong left quotient ringQls(R) of R and the strong left localization radicallRs are introduced and their properties are studied in detail. In particular, it is proved that Qls(Qls(R))≃Qls(R), lR/lRss=0 and a criterion is given for the ring Qls(R) to be a semisimple ring. There is a canonical homomorphism from the classical left quotient ring Ql,cl(R) to Qls(R) which is not an isomorphism, in general. The objects Qls(R) and lRs are explicitly described for several large classes of rings (semiprime left Goldie ring, left Artinian rings, rings with left Artinian left quotient ring, etc.).

New criteria for a ring to have a semisimple left quotient ring

Goldie's Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring. The aim of the paper is to give four new criteria (using a completely different approach and new ideas). The first one is based on the recent fact that for an arbitrary ring R the set ℳ of maximal left denominator sets of R is a non-empty set [V. V. Bavula, The largest left quotient ring of a ring, preprint (2011), arXiv:math.RA:1101.5107]: Theorem (The First Criterion). A ring R has a semisimple left quotient ring Q iff ℳ is a finite set, ⋂S∈ℳ ass (S) = 0 and, for each S ∈ ℳ, the ring S-1R is a simple left Artinian ring. In this case, Q ≃ ∏S∈ℳ S-1R. The Second Criterion is given via the minimal primes of R and goes further than the First one in the sense that it describes explicitly the maximal left denominator sets S via the minimal primes of R. The Third Criterion is close to Goldie's Criterion but it is easier to check in applications (basically, it reduces Goldie's Theorem to the prime case). The Fourth Criterion is given via certain left denominator sets.

Feckly Reduced Rings

Let R be a ring with identity and J(R) denote the Jacobson radical of R. In this paper, we introduce a new class of rings called feckly reduced rings. The ring R is called feckly reduced if R=J(R) is a reduced ring. We investigate relations between feckly reduced rings and other classes of rings. We obtain some characterizations of being a feckly reduced ring. It is proved that a ring R is feckly reduced if and only if every cyclic projective R-module has a feckly reduced endomorphism ring. Among others we show that every left Artinian ring is feckly reduced if and only if it is 2-primal, R is feckly reduced if and only if T(R;R) is feckly reduced if and only if R[x]= is feckly reduced.

ON REFINABLE MODULES

We study on cofinitely (strong) refinable modules as a proper generalization of (strongly) refinable modules. In this paper various properties of these modules are developed. It is shown that (1) every (cofinite) direct summand of a (cofinitely) refinable module is (cofinitely) refinable; (2) a module is cofinitely refinable if and only if every cofinite direct summands lift modulo every submodule of ; (3) a ring is Rad-supplemen-ted and left refinable if and only if it is semiperfect; (4) a semilocal ring is a left and right artinian serial ring with if and only if every left -module is strongly refinable. 2010 Mathematics Subject Classification. 16D10,16D40. Key words and phrases. (strongly) refinable module, cofinitely (strong) refinable module, semiperfect ring

Characterizations of left orders in left Artinian rings

Small (1966), Robson (1967), Tachikawa (1971) and Hajarnavis (1972) have given different criteria for a ring to have a left Artinian left quotient ring. In this paper, three more new criteria are given.