Nonsingular Rings Research Articles

The lattice coordinatized by a ring

A coordinatization functor L ( − , 1 ) : Ring → Latt is defined from the category of rings to the category of modular lattices. The main features of this coordinatization functor are 1) that it extends the functor R ↦ L ( R ) of von Neumann that associates to a regular ring its lattice of principal right ideals; 2) it respects the respective ( − ) op endofunctors on Ring and Latt ; and 3) it admits localization at a left R -module. The complemented elements of L ( R , 1 ) form a partially ordered set S ( R ) isomorphic to the space of direct summands of R R . The right nonsingular rings for which the embedding of S ( R ) into the localization L ( R , 1 ) Q at the right maximal ring of quotients Q R is an isomorphism are characterized by a property that every finite matrix subgroup φ ( R R ) of the left R -module R R is essential in an element of S ( R ) . In that case, the space S ( R ) obtains the structure of a complemented modular lattice coordinatized by the dominion, or equivalently, the ring of definable scalars, of the maximal ring of quotients. The class of rings with this property is elementary, in contrast to the class of rings whose space of right summands is coordinatized by the maximal ring of quotients.

Rings whose nonsingular right modules are $R$-projective

A right $R$-module $M$ is called $R$-projective provided that it is projective relative to the right $R$-module $R_{R}$. This paper deals with the rings whose all nonsingular right modules are $R$-projective. For a right nonsingular ring $R$, we prove that $R_{R}$ is of finite Goldie rank and all nonsingular right $R$-modules are $R$-projective if and only if $R$ is right finitely $\Sigma$-$CS$ and flat right $R$-modules are $R$-projective. Then, $R$-projectivity of the class of nonsingular injective right modules is also considered. Over right nonsingular rings of finite right Goldie rank, it is shown that $R$-projectivity of nonsingular injective right modules is equivalent to $R$-projectivity of the injective hull $E(R_{R})$. In this case, the injective hull $E(R_{R})$ has the decomposition $E(R_{R})=U_{R} \oplus V_{R}$, where $U$ is projective and $\operatorname{Hom}(V,R/I)=0$ for each right ideal $I$ of $R$. Finally, we focus on the right orthogonal class $\mathcal{N}^{\perp}$ of the class $\mathcal{N}$ of nonsingular right modules.

Derived decompositions of abelian categories I

Derived decompositions of abelian categories are introduced in internal terms of abelian subcategories to construct semi-orthogonal decompositions (or Bousfield localizations, or hereditary torsion pairs) in various derived categories of abelian categories. We give a sufficient condition for arbitrary abelian categories to have such derived decompositions and show that it is also necessary for abelian categories with enough projectives and injectives. For bounded derived categories, we describe which semi-orthogonal decompositions are determined by derived decompositions. The necessary and sufficient condition is then applied to the module categories of rings: localizing subcategories, homological ring epimorphisms, commutative noetherian rings and nonsingular rings. Moreover, for a commutative noetherian ring of Krull dimension at most $1$, a derived stratification of its module category is established.

Almost quasi clean rings

The element $q$ of a ring $R$ is called quasi-idempotent element if $q^2=uq$ for some central unit $u$ of $R$, or equivalently $q=ue$, where $u$ is a central unit and $e$ is an idempotent of $R$. In this paper, we define that the ring $R$ is almost quasi-clean if each element of $R$ is the sum of a regular element and a quasi-idempotent element. Several properties of almost-quasi clean rings are investigated. We prove that every quasi-continuous and nonsingular ring is almost quasi-clean. Finally, it is determined that the conditions under which the idealization of an $R$-module $M$ is almost quasi clean.

Divisibility and duo-rings

This paper investigates the projective dimension of the maximal right ring of quotients Q^r(R) of a right non-singular ring R. Our discussion addresses the question under which conditions pd (Q) \leq 1 guarantees that the module Q/R is a direct sum of countably generated modules extending Matlis’ Theorem for integral domains to a non-commutative setting.

On type-ADS modules and rings

In this paper, we introduce type absolute direct summand (type-ADS) modules and rings as a natural generalization of ADS ones. Besides basic properties and characterizations of the notion, we present several examples illustrating borders of the theory. We also show that some particular classical classes of rings, such as commutative or right non-singular rings are type-ADS.

Automorphism-invariant non-singular rings and modules

A ring A is a right automorphism-invariant right non-singular ring if and only if A=S×T, where S a right self-injective regular ring and T is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. Over a ring A, each direct sum of automorphism-invariant non-singular right modules is an automorphism-invariant module if and only if the factor ring of the ring A with respect to its right Goldie radical is a semiprime right Goldie ring.

Rings in which every ideal is pure-projective or FP-projective

A ring R is said to be left pure-hereditary (resp. RD-hereditary) if every left ideal of R is pure-projective (resp. RD-projective). In this paper, some properties and examples of these rings, which are nontrivial generalizations of hereditary rings, are given. For instance, we show that if R is a left RD-hereditary left nonsingular ring, then R is left Noetherian if and only if u.dim(RR)<∞. Also, we show that a ring R is quasi-Frobenius if and only if R is a left FGF, left coherent right pure-injective ring. A ring R is said to be left FP-hereditary if every left ideal of R is FP-projective. It is shown that if R is a left CF ring, then R is left Noetherian if and only if R is left pure-hereditary, if and only if R is left FP-hereditary, if and only if R is left coherent. It is shown that every left self-injective left FP-hereditary ring is semiperfect. Finally, it is shown that a ring R is left FP-hereditary (resp. left coherent) if and only if every submodule (resp. finitely generated submodule) of a projective left R-module is FP-projective, if and only if every pure factor module of an injective left R-module is injective (resp. FP-injective), if and only if for each FP-injective left R-module U, E(U)/U is injective (resp. FP-injective).

On the existence of maximal $\mathcal S$-closed submodules

The goal of this paper is to characterize the right non-singular rings R for which every non-singular right R -module contains a maximal \mathcal S -closed submodule. Several examples and related results are given.

Indecomposable decomposition and couniserial dimension

Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. We define and study couniserial dimension for modules. Couniserial dimension is a measure of how far a module deviates from being uniform. Despite their different objectives, it turns out that there are certain common properties between the couniserial dimension and Krull dimension. Among others, each module having such a dimension contains a uniform submodule and has finite uniform dimension. Like all dimensions, this is an ordinal valued invariant. Every module of finite length has couniserial dimension and its value lies between the uniform dimension and the length of the module. Modules with countable couniserial dimension are shown to possess indecomposable decomposition. In particular, a von Neumann regular ring with countable couniserial dimension is semisimple artinian. If the maximal right quotient ring of a semiprime right non-singular ring \(R\) has a couniserial dimension as an \(R\)-module, then \(R\) is a semiprime right Goldie ring. As one of the applications, it follows that all right \(R\)-modules have couniserial dimension if and only if \(R\) is a semisimple artinian ring.

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ⊗ R RG is a retractable RG-module if and only if M R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.

Rings and modules which are stable under automorphisms of their injective hulls

It is proved, among other results, that a prime right nonsingular ring (in particular, a simple ring) R is right self-injective if RR is invariant under automorphisms of its injective hull. This answers two questions raised by Singh and Srivastava, and Clark and Huynh. An example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption. Also shown is that automorphism-invariant modules are precisely pseudo-injective modules, answering a recent question of Lee and Zhou. Furthermore, rings whose cyclic modules are automorphism-invariant are investigated.

On Artinian rings with restricted class of injectivity domains

In a recent paper of Alahmadi, Alkan and López-Permouth, a ring R is defined to have no (simple) middle class if the injectivity domain of any (simple) R-module is the smallest or largest possible. Er, López-Permouth and Sökmez use this idea of restricting the class of injectivity domains to classify rings, and give a partial characterization of rings with no middle class. In this work, we continue the study of the property of having no (simple) middle class. We give a structural description of right Artinian right nonsingular rings with no right middle class. We also give a characterization of right Artinian rings that are not SI to have no middle class, which gives rise to a full characterization of rings with no middle class. Furthermore, we show that commutative rings with no middle class are those Artinian rings which decompose into a sum of a semisimple ring and a ring of composition length two. Also, Artinian rings with no simple middle class are characterized. We demonstrate our results with several examples.

RINGS WHICH ADMIT FAITHFUL TORSION MODULES II

Let R be an associative ring with identity. An (left) R-module M is said to be torsion if for every m ∈ M, there exists a nonzero r ∈ R such that rm = 0, and faithful provided rM = {0} implies r = 0 (r ∈ R). We call R (left) FT if R admits a nontrivial (left) faithful torsion module. In this paper, we continue the study of FT rings initiated in Oman and Schwiebert [Rings which admit faithful torsion modules, to appear in Commun. Algebra]. After presenting several examples, we consider the FT property within several well-studied classes of rings. In particular, we examine direct products of rings, Brown–McCoy semisimple rings, serial rings, and left nonsingular rings. Finally, we close the paper with a list of open problems.

RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES

AbstractDedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

Classes of Almost Clean Rings

A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and nonsingular module is almost clean and that every right CS (i.e. right extending) and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including finite AW *-algebras and noetherian Leavitt path algebras in particular, are almost clean. We say that a ring R is special clean (special almost clean) if each element a can be decomposed as the sum of a unit (regular element) u and an idempotent e with aR ∩ eR = 0. The Camillo-Khurana Theorem characterizes unit-regular rings as special clean rings. We prove an analogous theorem for abelian Rickart rings: an abelian ring is Rickart if and only if it is special almost clean. As a corollary, we show that a right quasi-continuous and right nonsingular ring is left and right Rickart. If a special (almost) clean decomposition is unique, we say that the ring is uniquely special (almost) clean. We show that (1) an abelian ring is unit-regular (equiv. special clean) if and only if it is uniquely special clean, and that (2) an abelian and right quasi-continuous ring is Rickart (equiv. special almost clean) if and only if it is uniquely special almost clean. Finally, we adapt some of our results to rings with involution: a *-ring is *-clean (almost *-clean) if each of its elements is the sum of a unit (regular element) and a projection (self-adjoint idempotent). A special (almost) *-clean ring is similarly defined by replacing “idempotent” with “projection” in the appropriate definition. We show that an abelian *-ring is a Rickart *-ring if and only if it is special almost *-clean, and that an abelian *-ring is *-regular if and only if it is special *-clean.

Characterization of Some Ring Properties in Incidence Algebras

Let R be a ring with identity and I(X, R) be the incidence algebra of a locally finite partially ordered set X over R. In this article, we investigate the necessary and sufficient conditions for the incidence ring to be Ikeda-Nakayama, nil injective, NI, reduced, nonsingular and Kasch ring.

Finitely Presented Modules over Right Non-Singular Rings

This paper charactyerizes the right non-singular rings R for which M/Z(M) is projective whenever M is a cyclically (finitely) presented module. Several related results investigate right semi-hereditary rings.

Quasi Co-Hopfian Modules and Applications

We carry out an extensive study of modules M R with the property that M/f(M) is singular for all injective endomorphisms f of M. Such modules called “quasi co-Hopfian”, generalize co-Hopfian modules. It is shown that a ring R is semisimple if and only if every quasi co-Hopfian R-module is co-Hopfian. Every module contains a unique largest fully invariant quasi co-Hopfian submodule. This submodule is determined for some modules including the semisimple ones. Over right nonsingular rings several equivalent conditions to being quasi co-Hopfian are given. Modules with all submodules quasi co-Hopfian are called “completely quasi co-Hopfian” (cqcH). Over right nonsingular rings and over certain right Noetherian rings, it is proved that every finite reduced rank module is cqcH. For a right nonsingular ring which is right semi-Artinian (resp. right FBN) the class of cqcH modules is the same as the class of finite reduced rank modules if and only if there are only finitely many isomorphism classes of nonsingular R-modules which are simple (resp. indecomposable injective).

ESSENTIALLY RETRACTABLE MODULES

We call a module R M essentially retractable if HomR( ) ,0 MN ≠ for all essential submodules N of M. For a right FBN ring R, it is shown that: (i) A nonzero module R M is retractable (in the sense that HomR( ) ,0 MN ≠ for all nonzero R M N ≤ ) if and only if certain factor modules of M are essentially retractable nonsingular modules over R modulo their annihilators. (ii) A non-zero module R M is essentially retractable if and only if there exists a prime ideal () R M Ass P∈ such that HomR( ) ,0 MN ≠ . Over semiprime right nonsingular rings, a nonsingular essentially retractable module is precisely a module with nonzero dual. Moreover, over certain rings R, including right FBN rings, it is shown that a nonsingular module M with enough uniforms is essentially retractable if and only if there exist uniform retractable R-modules { } I i i U ∈ and R-homomorphisms M U M i I i ⎯→ ⎯ ⊕ ⎯→ ⎯ ∈ β α with 0 βα ≠ .