Semiperfect Rings Research Articles

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.

A Study On a New Generalization of $\delta$-Supplemented Modules

For any ring $S$ and an $S$-module $W$, a submodule $G$ of $W$ is termed \emph{co$_\delta$-coatomic} if the quotient module $W/G$ is $\delta$-coatomic. In this study, we introduce the term ($\oplus$-)\emph{co$_\delta$-coatomically $\delta$-supplemented module}, or shortly ($\oplus$-)\emph{co$_\delta$-$\delta$-supplemented module} to describe a module $W$ where each co$_\delta$-coatomic submodule has a $\delta$-supplement (that is a direct summand) in $W$. Furthermore, a module $W$ is identified as \emph{co$_\delta$-coatomically $\delta$-semiperfect}, or shortly \emph{co$_\delta$-$\delta$-semiperfect}, provided each $\delta$-coatomic quotient module of $W$ has a projective $\delta$-cover. It has been proved that over a $\delta$-semiperfect ring $S$, the module $_{S}S$ is $\oplus_{\delta}$-co-coatomically supplemented if and only if $_{S}S$ is co$_\delta$-$\delta$-semiperfect if and only if $_{S}S$ is $\oplus$-co$_\delta$-$\delta$-supplemented.

Homological dimensions of the Jacobson radical

This work presents results on the finiteness, and on the symmetry properties, of various homological dimensions associated to the Jacobson radical and its higher syzygies, of a semiperfect ring.

Rings Characterizations with Mutually SS-Supplemented Modules

In this text, the notion of mutually ss-supplemented modules characterizes with the help of semiperfect rings. For this mutually ss-supplemented modules were first classified according to certain properties. These features can be listed as follows in the refinable modules, distributive modules, fully invariant submodules and (π-)projective modules. Especially it was proven that each 𝜋-projective mutually ss-supplemented module is amply mutually ss-supplemented. It was obtained that every ss-supplemented refinable module has direct summand which is a mutually ss-supplement in the module. It was shown that C=⨁_(ϱ∈Λ ) C_ϱ is a mutually ss-supplemented module which each submodule of C is a fully invariant submodule, for the family of mutually ss-supplemented modules {C_ϱ }_(ϱ∈Λ).

Matrices having nonzero outer inverses

It is well known that every nonzero von Neumann regular $m\times n$-matrix $A$ over an arbitrary ring $R$ has a nonzero outer inverse $n\times m$-matrix $B$ in the sense that $B=BAB$. Generalizing previous work on von Neumann regular matrices, the matrices having nonzero outer inverses over semiperfect rings are characterized as the matrices having some entry outside the Jacobson radical of $R$. Such matrices over finite semiperfect rings and finite commutative rings are counted, and several applications are given.

Generalization of ⊕-Cofinitely Radical Supplemented Modules

In this study, we clarify 𝑚𝑔𝑠⊕−modules that are the generalization of ⊕−cofinitely radical supplemented modules and look at some of their basic characteristics. Additionally, we determine the prerequisites for the factor module of an arbitrary 𝑚𝑔𝑠⊕−module to be a 𝑚𝑔𝑠⊕−module and characterized semiperfect rings with the aid of this module.

A look at FPF rings

An error in Corollary 9.32 of [C. Faith, Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Mathematical Surveys and Monographs, Vol. 65 (American Mathematical Society, Providence, RI, 2004)], motivated us to consider again FPF rings which were initiated by Faith in the 1970s. In this paper, it is shown that a commutative ring [Formula: see text] is reduced FPF if and only if it is [Formula: see text]-semihereditary. We show that when a semiperfect ring with a strongly right bounded basic ring with right and left Ore conditions, is an FPF ring. After some general results, the article focuses on rings of continuous functions. We give some algebraic characterizations for a [Formula: see text] to be FPF and retrieve a result of Jorge Martinez. Also, we show that a space [Formula: see text] is fraction-dense if and only if [Formula: see text] is a continuous ring.

Relation between balanced pairs and TTF triples

Let A be a complete and cocomplete abelian category with the additional assumption that any direct sum and product of short exact sequences are exact. We explore the relation between balanced pairs and TTF triples in A. The main results are: (1) every balanced pair in A induces a TTF triple; (2) if A has projective covers and injective envelopes, then every TTF triple in A gives rise to a balanced pair, and hence there is a bijection between the equivalence classes of balanced pairs and TTF triples in A; (3) a balanced pair in A is quasi admissible if and only if its induced TTF triple is centrally splitting. Our first application of these results provide abundant rings over which every balanced pair is quasi admissible, including local rings, commutative semiperfect rings, and commutative Noetherian rings. Another application is the classification of equivalence classes of cohereditary balanced pairs over arbitrary rings. We also present counterexamples to [15, Open questions 3 and 5]. Finally, we prove that the answers to [15, Open questions 2 and 4] are positive for coherent rings with weak global dimension at most one.

Von Neumann local matrices

We use our recent results on von Neumann regular matrices, strongly regular matrices and matrices having a non-zero outer inverse to derive applications to some generalizations of these concepts, called von Neumann local, strongly von Neumann local and outer von Neumann local matrices. Among other properties, we show that the $t^{\rm th}$ compound matrix of every matrix of determinantal rank $t$ over a commutative local ring is strongly von Neumann local, and every matrix over an arbitrary semiperfect ring is outer von Neumann local.

On the automorphism-invariance of finitely generated ideals and formal matrix rings

In this paper, we study rings having the property that every finitely generated right ideal is automorphism-invariant. Such rings are called right f a-rings. It is shown that a right f a-ring with finite Goldie dimension is a direct sum of a semisimple artinian ring and a basic semiperfect ring. Assume that R is a right f a-ring with finite Goldie dimension such that every minimal right ideal is a right annihilator, its right socle is essential in RR, R is also indecomposable (as a ring), not simple, and R has no trivial idempotents. Then R is QF. In this case, QF-rings are the same as q?, f q?, a?, f a-rings. We also obtain that a right module (X,Y, f, g) over a formal matrix ring (R M N S) with canonical isomorphisms f? and g? is automorphism-invariant if and only if X is an automorphism-invariant right R-module and Y is an automorphism-invariant right S-module.

Generalized Koszul algebra and Koszul duality

We define generalized Koszul modules and rings and develop a generalized Koszul theory for N-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings with degree zero part artinian semisimple developed by Beilinson-Ginzburg-Soergel and the ungraded Koszul theory for noetherian semiperfect rings developed by Green and Martinéz-Villa. Let A be a left finite N-graded ring generated in degree 1 with A0 noetherian semiperfect, J be its graded Jacobson radical. By the Koszul dual of A we mean the Yoneda Ext ring Ext_A•(A/J,A/J). If A is a generalized Koszul ring and M is a generalized Koszul module, then it is proved that the Koszul dual of the Koszul dual of A is the associated graded ring GrJA and the Koszul dual of the Koszul dual of M is the associated graded module GrJM. If A is a locally finite algebra, then the following statements are proved to be equivalent: A is generalized Koszul; the Koszul dual Ext_A•(A/J,A/J) of A is (classically) Koszul; GrJA is (classically) Koszul; the opposite ring Aop of A is generalized Koszul. As an application, it is proved that if A is generalized Koszul with finite global dimension then A is generalized AS regular if and only if the Koszul dual of A is self-injective.

Symmetry on zero and idempotents

Alghazzawi and Leroy studied the structure of subsets satisfying the properties of symmetric and commutatively closed, that is, for implies and for implies respectively, where S is a subset of a ring R. In this article we discuss the structure of rings which are symmetric on zero (resp., idempotents). Such rings are also called symmetric (resp., I-symmetric). We first prove that if a polynomial over a symmetric ring is a unit then a 0 is a unit and ai is nilpotent for all based on this result, we obtain that for a reduced ring R, the group of all units of the polynomial ring over R coincides with one of R, and that polynomial rings over I-symmetric rings are identity-symmetric. It is proved that for an abelian semiperfect ring R, R is I-symmetric if and only if the units in R form an Abelian group if and only if R is commutative. It is also proved that for an I-symmetric ring R, R is π-regular if and only if is a commutative regular ring and J(R) is nil, where J(R) is the Jacobson radical of R.

The isomorphism problem for graph magma algebras

(One-value) graph magma algebras are algebras having a basis such that, for all Such bases induce graphs and, conversely, certain types of graphs induce graph magma algebras. The equivalence relation on graphs that induce isomorphic magma algebras is fully characterized for the class of associative graphs having only finitely many non-null connected components. In the process, the ring-theoretic structure of the magma algebras induced by those graphs is given as it is shown that they are precisely those graph magma algebras that are semiperfect as rings. A complete description of the semiperfect rings that arise in this fashion, in ring theoretic and linear algebra terms, is also given. In particular, the precise number of isomorphism classes of one-value magma algebras of dimension n is shown to be where, for any p(i) is the number of partitions of i. While it is unknown whether uncountable dimensional algebras always have amenable bases, it is shown here that graph magma algebras do.

The depth, the delooping level and the finitistic dimension

We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call the “delooping level”. These give lower and upper bounds for the finitistic dimension, respectively. As first theorems, we give a necessary and sufficient criterion for the delooping level to be finite in terms of the splitting of the unit map of a related adjunction, and use this to give a sufficient torsionfreeness criterion for finiteness. We further relate these invariants to the Auslander–Bridger grade conditions for modules, which are vanishing conditions on double Ext duals. As main theorem, we prove that these bounds agree whenever the first non-trivial grade conditions are satisfied for simple modules, so that either invariant computes the finitistic dimension in this case. Over Artinian rings, we show that the delooping level also bounds the big finitistic dimension, and we obtain a sufficient cohomological criterion for the first finitistic dimension conjecture to hold. Over commutative local Noetherian rings, these conditions always hold and we obtain a new characterisation of the depth as the delooping level of the ring.

The Class of Noetherian RingsWith Finite Valuation Dimension

Not a long time ago, Ghorbani and Nazemian [2015] introduced the concept of dimension of valuation which measures how much does the ring differ from the valuation. They've shown that every Artinian ring has a finite valuation dimensions. Further, any comutative ring with a finite valuation dimension is semiperfect. However, there is a semiperfect ring which has an infinite valuation dimension. With those facts, it is of interest to further investigate property of rings that has a finite dimension of valuation. In this article we define conditions that a Noetherian ring requires and suffices to have a finite valuation dimension. In particular we prove that, if and only if it is Artinian or valuation, a Noetherian ring has its finite valuation dimension. In view of the fact that a ring needs a semi perfect dimension in terms of valuation, our investigation is confined on semiperfect Noetherian rings. Furthermore, as a finite product of local rings is a semi perfect ring, the inquiry into our outcome is divided into two cases, the case of the examined ring being local and the case where the investigated ring is a product of at least two local rings. This is, first of all, that every local Noetherian ring possesses a finite valuation dimension, if and only if it is Artinian or valuation. Secondly, any Notherian Ring generated by two or more local rings is shown to have a finite valuation dimension, if and only if it is an Artinian.

Generalized “stacked bases” theorem for modules over semiperfect rings

The history of generalized “stacked bases” theorem origins from the result of Hill and Megibben on abelian groups. We extend this theorem for modules over semiperfect rings and as a consequence we show that for a submodule H of a projective module G over a semiperfect ring, the following conditions are equivalent: there exists a decomposition into a direct sum of indecomposable modules Pi , such that G/H is a direct sum of a family of modules, isomorphic to factor modules of principal indecomposable modules.

SS-discrete modules

In this paper, we define (strongly) ss-discrete, semi-ss-discrete and quasi-ss-discrete modules as a strongly notion of (strongly) discrete, semi-discrete and quasi-discrete modules with the help of ss-supplements in [3]. We examined the basic properties of these modules and included characterization of strongly ss-discrete modules over semi-perfect rings.

Peirce decompositions, idempotents and rings

Idempotents dominate the structure theory of rings. The Peirce decomposition induced by an idempotent provides a natural environment for defining and classifying new types of rings. This point of view offers a way to unify and to expand the classical theory of semiperfect rings and idempotents to much larger classes of rings. Examples and applications are included.

Volodymyr Vasylovych Kyrychenko

The paper is dedicated to the memory of an outstanding ukrainian mathematician and teacher, one of the founders of Kyiv school of representation theory and ring theory, professor of Taras Shevchenko National University of Kyiv Volodymyr Vasylovych Kyrychenko. Volodymyr Vasylovych was born in 1942 in Penza. He graduated Kyiv University with honors in 1964. He defended his PhD thesis “Representations of hereditary, completely decomposable and Bass orders” under the supervision of D.K. Faddeev. He defended his habilitation “Modules and structural theory of rings” in 1986. He started to work at Kyiv University as an assistant of the Algebra and Mathematical Logic Department in 1967. Since then he served as an Associate Professor and Professor of the Algebra and Mathematical Logic Department. He was the head of the Geometry Department from 1988 till 2016. His main scientific results lay in the field of structural ring theory. In particular, he introduced the notion of the quiver for some special casses of rings and developed the structural theory of hereditary semiserial semidistributive rings. He is the author of about 250 scientific papers and 5 fundamental monographies. In his monographies the theory of semiperfect rings is presented by means of correspond- ing quivers. The monography “Finite dimensional algebras” written jointly with professor Y. Drozd is regarded one of the classical works in algebra and translated from Russian to Chineese, English and Spanish. He was an outstanding teacher and 36 PhD thesis were defended under his supervision. His lectures on analytical geometry and ring theory were always top ranked among students. Professor Kyrychenko was one of the founders of the journal “Algebra and Discrete Mathematics”. He was an editor of a couple of mathematical journals. In 1997 he was one of the founders of the series of international algebraic conferences in Ukraine that were conducted every two years. Since that he was one of the main organizers of each of them.

Extended S-supplement submodules

In this paper, we introduced and studied extended S-supplement submodules. A submodule U of a module V is called extended S-supplement submodule in V if there exists a submodule T of V such that V = T + U and U ∩ T is Goldie torsion. Extended S-supplement submodule is a dual notion of extended S-closed submodule. The class of extended S-supplement sequences is a proper class which is generated by nonsingular modules injectively. We studied coinjective objects of this class. Moreover, extended S-supplemented modules are also investigated. We present new characterizations of Z2(RR) -semiperfect rings and SI-rings by extended S-supplement submodules.