Goldie Dimension Research Articles

On iso-DICC modules

In this paper, the concept of iso-DICC modules is introduced and extensively investigated. These modules are the generalization of DICC, iso-Noetherian and iso-Artinian modules. We prove that any direct sum of a projective iso-simple and an iso-DICC module is iso-DICC. Unlike DICC rings, we show that iso-DICC rings don’t have finite Goldie dimension, but if R is a semiprime right iso-DICC ring and I is an annihilator ideal of R, then either I or R I has finite Goldie dimension. Furthermore, we investigate the Krull dimension of iso-Noetherian modules and we give an answer to an open problem on this topic. Finally, we generalize some results on semiprime right iso-Artinian rings to a certain class of iso-Artinian modules. We provide sufficient conditions for iso-Artinian modules to have finite Goldie dimension. Also, we show that any semiprime FQS iso-Artinian module, is non-M-singular.

Goldie dimension for C*-algebras

AbstractIn this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.

On the Goldie dimension of finitely generated locally cyclic modules

Let R be a commutative ring with identity. In this paper we investigate the Goldie dimension of finitely generated locally cyclic R-modules. Then, we give a characterization of rings whose finitely generated locally cyclics have finite Goldie dimension.

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.

On the finite Goldie dimension of sum of two ideals of an R-group

On the finite Goldie dimension of sum of two ideals of an R-group

On parallel Krull dimension of modules

In this article, we introduce the concept of parallel Krull dimension of a module (briefly, p.Krull dimension), which is Krull-like dimension extension of the concept of DCC on poset of submodules parallel to itself. Using this concept, we extend some basic results about modules with this dimension. In particular, we show that an R-module M has Krull dimension if and only if it has homogeneous parallel Krull dimension with finite Goldie dimension and these two dimensions for M coincide. Furthermore, we define the concept of α-DICCP and we prove that M is an α-DICC module if and only if M is an α-DICCP module with finite Goldie dimension. Also, after defining of p-Artinian (resp., p-Noetherian) modules, we prove that if R is semiprime, p-Artinian commutative ring with finite Goldie dimension, then R is a Goldie ring.

Modules with chain condition on uncountably generated submoduled

In this paper, we study modules with chain condition on uncountably generated submodules. We show that if an [Formula: see text]-module [Formula: see text] satisfies the ascending chain condition on uncountably generated submodules, then its Goldie dimension is less than or equal to [Formula: see text], where [Formula: see text] is the first uncountable cardinal number. We also show that if a quotient finite dimensional module [Formula: see text] satisfies the ascending chain condition on uncountably generated submodules, then it has Noetherian dimension and its Noetherian dimension is less than or equal to [Formula: see text], where [Formula: see text] is the first uncountable ordinal number. We also investigate that if a quotient finite dimensional module [Formula: see text] satisfies the descending chain condition on uncountably generated submodules, then [Formula: see text] has Krull dimension and its Krull dimension may be any ordinal number [Formula: see text].

The Zariski covering number for vector spaces and modules

Given a module M over a commutative unital ring R, let denote the covering number, i.e. the smallest (cardinal) number of proper submodules whose union covers M; this includes the covering numbers of Abelian groups, which are extensively studied in the literature. Recently, Khare–Tikaradze [Comm. Algebra, in press] showed in several cases that where SM is the set of maximal ideals with Our first main result extends this equality to all R-modules with small Jacobson radical and finite dual Goldie dimension. We next introduce and study a topological counterpart for finitely generated R-modules M over rings R, whose ‘some’ residue fields are infinite, which we call the Zariski covering number To do so, we first define the “induced Zariski topology” τ on M, and now define to be the smallest (cardinal) number of proper τ-closed subsets of M whose union covers M. We then show our next main result: for all finitely generated R-modules M for which (a) the dual Goldie dimension is finite, and (b) whenever is finite. As a corollary, this alternately recovers the aforementioned formula for the covering number of the aforementioned finitely generated modules. Finally, we discuss the notion of κ-Baire spaces, and show that the inequalities again become equalities when the image of M under the continuous map (with appropriate Zariski-type topologies) is a κM -Baire subspace of the product space.

Pure Goldie Dimensions for Exactly Definable Categories

It is shown that if A is an object in an exactly definable category $${\mathcal {C}}$$ such that A has finite pure Goldie dimension and that every pure monomorphism $$A\rightarrow A$$ is an isomorphism, then its endomorphism ring $$\hbox {End}_{{\mathcal {C}}}(A)$$ is semilocal. Also, it is proved that every subobject of a pure quotient finite dimensional pure injective object of an exactly definable category has a semilocal endomorphism ring.

Distributive lattices with strong endomorphism kernel property as direct sums

In this paper,for a distributive lattice $mathcal L$, we study and compare some lattice theoretic features of $mathcal L$ and topological properties of the Stone spaces ${rm Spec}(mathcal L)$ and ${rm Max}(mathcal L)$ with the corresponding graph theoretical aspects of the zero-divisor graph $Gamma(mathcal L)$.Among other things,we show that the Goldie dimension of $mathcal L$ is equal to the cellularity of the topological space ${rm Spec}(mathcal L)$ which is also equal to the clique number of the zero-divisor graph $Gamma(mathcal L)$. Moreover, the domination number of $Gamma(mathcal L)$ will be compared with the density and the weight of the topological space ${rm Spec}(mathcal L)$. For a $0$-distributive lattice $mathcal L$, we investigate the compressed subgraph $Gamma_E(mathcal L)$ of the zero-divisor graph $Gamma(mathcal L)$ and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of $mathcal L$.

CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES

We introduce the notion of locally supplemented modules (i.e., modules for which every finitely generated submodule is supplemented). We show that a module $M$ is locally supplemented if and only if $M$ is a sum of local submodules. We characterize several classes of rings in terms of locally supplemented modules. Among others, we prove that a ring $R$ is a Camillo ring if and only if every finitely embedded $R$-module is locally supplemented. It is also shown that a ring $R$ is a Gelfand ring if and only if every $R$-module having a finite Goldie dimension is locally supplemented.

Homomorphisms with Semilocal Endomorphism Rings Between Modules

We study the category Morph(Mod-R) whose objects are all morphisms between two right R-modules. The behavior of the objects of Mod-R whose endomorphism ring in Morph(Mod-R) is semilocal is very similar to the behavior of modules with a semilocal endomorphism ring. For instance, direct-sum decompositions of a direct sum $\oplus _{i=1}^{n}M_{i}$ , that is, block-diagonal decompositions, where each object Mi of Morph(Mod-R) denotes a morphism $\mu _{M_{i}}\colon M_{0,i}\to M_{1,i}$ and where all the modules Mj,i have a local endomorphism ring End(Mj,i), depend on two invariants. This behavior is very similar to that of direct-sum decompositions of serial modules of finite Goldie dimension, which also depend on two invariants (monogeny class and epigeny class). When all the modules Mj,i are uniserial modules, the direct-sum decompositions (block-diagonal decompositions) of a direct-sum $\oplus _{i=1}^{n}M_{i}$ depend on four invariants.

Dimension on non-essential submodules

In this paper, we introduce and study the concepts of non-essential Krull dimension and non-essential Noetherian dimension of an [Formula: see text]-module, where [Formula: see text] is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension. They respectively rely on the behavior of descending and ascending chains of non-essential submodules. It is proved that each module with non-essential Krull dimension (respectively, non-essential Noetherian dimension) has finite Goldie dimension. We also show that a semiprime ring [Formula: see text] with non-essential Noetherian dimension is uniform.

On fusible rings

We answer in negative two of questions posed in Ghashghaei and McGovern. We also establish a new characterization of semiprime left Goldie rings by showing that a semiprime ring R is left Goldie iff it is regular left fusible and has finite left Goldie dimension.

The Krull-Schmidt Theorem Holds for Finite Direct Products of Biuniform Groups

We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups G whose lattice of normal subgroups $\mathcal {N}(G)$ has Goldie dimension and dual Goldie dimension 1. More generally, it holds for the class of completely indecomposable groups.

Algebraic properties of some factor rings of C(X)

In F. Azarpanah et al. (2008) [3] the authors have given some algebraic properties of the ring C(X)/CF(X), where CF(X)=OβX∖I(X). In this paper, first, we show that C(X)/CF(X) is a C-ring if and only if the set of isolated points of X is finite. Next, we generalize this work for rings C(X)/OA and C(X)/MA whenever A⊆βX (or just a closed one, in some cases) and then topological conditions on A for which every prime (maximal) ideal of C(X)/OA (resp., C(X)/MA) is essential are characterized. We call a ring R an EIN-ring if for each two orthogonal ideals I, J of R which are generated by two subsets of idempotents, Ann(I)+Ann(J)=R. It is shown for a closed subset A of βX that C(X)/OA is an EIN-ring if and only if C(X)/MA is an EIN-ring if and only if A is an EF-space. Minimal ideals, socle and the intersection of all essential maximal ideals of C(X)/OA (resp., C(X)/MA) are characterized. We prove also that dim(C(X)/OA)≥dimC(X)/MA, where dimC(X)/MA denotes the Goldie dimension of C(X)/MA, and the inequality may be strict.

On pure Goldie dimensions

ABSTRACTIn this paper, we examine the pure Goldie dimension and dual pure Goldie dimension in finitely accessible additive categories. In particular, we show that if A is an object in a finitely accessible additive category 𝒜 that has finite pure Goldie dimension n and finite dual pure Goldie dimension m, then End𝒜(A) is semilocal and the dual Goldie dimension of End𝒜(A) is less than or equal to n+m.

A note on semiprime right Goldie rings

ABSTRACTIt is shown that a ring R is semiprime right Goldie if and only if R is right nonsingular and every nonsingular right R-module M has a direct decomposition M = I⊕N, where I is injective and N is a reduced module such that N does not contain any extending submodule of infinite Goldie dimension.

On Objects with a Semilocal Endomorphism Rings in Finitely Accessible Additive Categories

It is proved that if A is an object in a finitely accessible additive category $\mathcal {A}$ such that A has finite pure Goldie dimension and that every pure monomorphism A→A is an isomorphism, then its endomorphism ring $End_{\mathcal {A}}(A)$ is semilocal.

Avoiding modules, co-avoiding modules, Goldie dimension and its dual

We introduce and study two concepts in module theory which are dual to each other. It will be revealed that, in the finite case, they are new equivalent concepts for the Goldie dimension of modules and the dual Goldie dimension. However, we shall observe, in the infinite case, they are different concepts.