Goldie Dimension Research Articles

The zero-divisor graph of a reduced ring

In this paper the zero-divisor graph Γ ( R ) of a commutative reduced ring R is studied. We associate the ring properties of R , the graph properties of Γ ( R ) and the topological properties of Spec ( R ) . Cycles in Γ ( R ) are investigated and an algebraic and a topological characterization is given for the graph Γ ( R ) to be triangulated or hypertriangulated. We show that the clique number of Γ ( R ) , the cellularity of Spec ( R ) and the Goldie dimension of R coincide. We prove that when R has the annihilator condition and 2 ∉ Z ( R ) ; Γ ( R ) is complemented if and only if Min ( R ) is compact. In a semiprimitive Gelfand ring, it turns out that the dominating number of Γ ( R ) is between the density and the weight of Spec ( R ) . We show that Γ ( R ) is not triangulated and the set of centers of Γ ( R ) is a dominating set if and only if the set of isolated points of Spec ( R ) is dense in Spec ( R ) .

On the Goldie dimension of rings and modules

We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generating sets of their quasi-injective hulls. Several consequences are deduced. In particular, it is shown that every finitely generated hereditary module with countably generated quasi-injective hull is noetherian. It is also shown that every right hereditary ring with finitely generated injective hull is right artinian, thus answering a long standing open question posed by Dung, Gómez Pardo and Wisbauer.

ARTINIANNESS OF LOCAL COHOMOLOGY MODULES OF ZD-MODULES

This article centers around artinianness of the local cohomology of ZD-modules. Let a he an ideal of a commutative Noetherian ring R. The notion of a-relative Goldie dimension of an R-module M, as a generalization of that of Goldie dimension is presented. Let M be a ZD-module, such that the a-relative Goldie dimension of any quotient of M is finite. It is shown that if dim R/a = 0, then the local cohomology modules Hi a(M) are artinian. Also, it is proved that if d = dim M is finite, then Hd a (M) is artinian, for any ideal a of R. These results extend the previously-known results concerning artinianness of local cohomology of finitely generated modules.

Zero-divisor graph of C(X)

In this article the zero-divisor graph Γ(C(X)) of the ring C(X) is studied. We associate the ring properties of C(X), the graph properties of Γ(C(X)) and the topological properties of X. Cycles in Γ(C(X)) are investigated and an algebraic and a topological characterization is given for the graph Γ(C(X)) to be triangulated or hypertriangulated. We have shown that the clique number of Γ(C(X)), the cellularity of X and the Goldie dimension of C(X) coincide. It turns out that the dominating number of Γ(C(X)) is between the density and the weight of X. Finally we have shown that Γ(C(X)) is not triangulated and the set of centers of Γ(C(X)) is a dominating set if and only if the set of isolated points of X is dense in X if and only if the socle of C(X) is an essential ideal.

COUNIFORM DIMENSION OVER SKEW POLYNOMIAL RINGS#

ABSTRACT In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism σ ∈ Aut(R), we use the notion of a σ-compatible module M R to obtain results on the couniform dimension of the polynomial modules M[x], M[x −1], and M[x, x −1] over suitable skew extension rings.

On Some Properties of Essential Darboux Rings of Real Functions Defined on Topological Spaces

This paper deals with rings of real Darboux functions defined on some topological spaces. Results are given concerning the existence of essential, as well as prime Darboux rings. We prove that, under some assumptions connected with the domain $X$ of the functions, the equalities: $D(X)=S_{lf}(X), S(X)=\dim(\Re )$ hold, where $D(X)$ is a ${\cal D}$-number of $X$, $S(X)$ ($S_{lf}(X)$) denotes the Souslin (lf-Souslin) number of $X$ and $\dim(\Re )$ is a Goldie dimension of an arbitrary prime Darboux ring $\Re$.

Artinian Rings Characterized by Direct Sums of CS Modules#

In this note, we show that the following are equivalent for a ring R for which the socle or the injective hull of R R is finitely generated: (i) The direct sum of any two CS right R-modules is again CS; (ii) R is right Artinian and every uniform right R-module has composition length at most two. Next we give partial answers to a question of Huynh whether a right countably Σ-CS ring which either is semilocal or has finite Goldie dimension is right Σ-CS. We give characterizations, in terms of radicals, of when such rings are right Σ-CS. In particular, for the semilocal case, Huynh's question is reduced to whether rad(Z 2(R R )) is Σ-CS or Noetherian, where Z 2(R R ) is the second singular right ideal of R. Our results yield new characterizations of QF-rings.

Left Quotient Associative Pairs and Morita Invariant Properties#

In this paper, we prove that left nonsingularity and left nonsingularity plus finite left local Goldie dimension are two Morita invariant properties for idempotent rings without total left or right zero divisors. Moreover, two Morita equivalent idempotent rings, semiprime and left local Goldie, have Fountain–Gould left quotient rings that are Morita equivalent too. These results can be obtained from others concerning associative pairs. We introduce the notion of (general) left quotient pair of an associative pair and show the existence of a maximal left quotient pair for every semiprime or left nonsingular associative pair. Moreover, we characterize those associative pairs for which their maximal left quotient pair is von Neumann regular and give a Gabriel-like characterization of associative pairs whose maximal left quotient pair is semiprime and artinian.

Two Results on Modules whose Endomorphism Ring is Semilocal

The aim of this paper is twofold. On the one hand, we show that the dual Goldie dimension codim(End(MR)) of the endomorphism ring End(MR) of a module MR can be used as a measure of the dimension of the module MR. On the other hand, we prove under suitable hypotheses the validity of the Krull–Schmidt Theorem for infinite direct sums of modules with homogeneous semilocal endomorphism rings.

Weak Krull–Schmidt theorem and direct sum decompositions of serial modules of finite Goldie dimension

We show that every direct summand of a serial module M of finite Goldie dimension is serial, give a description of the corresponding commutative monoid V ( M ) , and generalize Facchini's weak Krull–Schmidt theorem to a larger class of modules.

Modules with RD-Composition Series over a Commutative Ring

If R is a commutative ring,then we prove that every finitely generated R-module has a pure-composition series with indecomposable cyclic factors and any two such series are isomorphic if and only if R is a Bézout ring and a CF-ring. When R is a such ring,the length of a pure-composition series of a finitely generated R-module M is compared with its Goldie dimension and we prove that these numbers are equal if and only if M is a direct sum of cyclic modules. We also give an example of an artinian module over a noetherian domain,which has an RD-composition series with uniserial factors. Finally we prove that every pure-injective R-module is RD-injective if and only if R is an arithmetic ring.

Modules with only Finitely Many Direct Sum Decompositions up to Isomorphism

In this paper we study almost Krull-Schmidt modules, that is, modules with only finitely many direct sum decompositions up to isomorphism. This is a finiteness con- dition on modules that has nothing to do with the other finiteness conditions usually considered in the mathematical literature, like being noetherian, or AB5 ⁄ , or having finite Goldie dimension, or finite dual Goldie dimension, or finite Krull dimension. Then we compute bounds on the number and the lengths of the direct sum decompositions of modules.

On some properties of ⊕‐supplemented modules

A module M is ⊕‐supplemented if every submodule of M has a supplement which is a direct summand of M. In this paper, we show that a quotient of a ⊕‐supplemented module is not in general ⊕‐supplemented. We prove that over a commutative ring R, every finitely generated ⊕‐supplemented R‐module M having dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ring R is semisimple if and only if the class of ⊕‐supplemented R‐modules coincides with the class of injective R‐modules. The structure of ⊕‐supplemented modules over a commutative principal ideal ring is completely determined.

$u$-Independence and Quadratic $u$-Independence in the Construction of Indecomposable Finitely Generated Modules

Let R be a valuation domain having an ideal I such that a maximal immediate extension S of R contains four units u-independent over I. We construct a 4-generated indecomposable R-module M with Goldie dimension g(M) = 2. We thus supplement a result by Lunsford who constructed indecomposable finitely generated R-modules making use of sets of quadratically u-independent elements of S.

Generalized Hopfian modules

A module is called generalized Hopfian (gH) if any of its surjective endomorphisms has a small kernel. Such modules are in a sense dual to weakly co-Hopfian modules that were defined and extensively studied in [A. Haghany, M.R. Vedadi, J. Algebra 243 (2001) 765–779]. Several equivalent formulations are given for gH modules and used in their study. Generalized Hopfian modules form a proper subclass of Dedekind finite modules, but this subclass is rather large in that it properly contains Hopfian, and in particular Noetherian modules, as well as Artinian modules and modules with finite dual Goldie dimension. For quasi-projective modules, the properties Dedekind finite, Hopfian and gH are all equivalent, yielding a ring R stably finite iff all finitely generated free R modules are gH. The gH property is shown to be a Morita invariant, and rings are characterized over which all finitely generated modules are gH. Some other classes of rings are also studied. These rings satisfy one of the conditions: all right ideals are gH; all annihilator right ideals are small (right domain); all left regular elements have small right annihilators (right directly finite). We will show: right domain ⇒ right directly finite ⇒ directly finite and these cannot be reversed. These results are by products of the study of special gH modules.

On representable linearly compact modules

For a flat R-module F, we prove that Hom R (F, -) is a functor from the category of linearly compact R-modules to itself and is exact. Moreover, Hom R (F, M) is representable when M is linearly compact and representable. This gives an affirmative answer to a question of L. Melkersson (1995) for linearly compact modules without the condition of finite Goldie dimension. The set of attached prime ideals of the co-localization Hom R (R S , M) of a linearly compact representable R-module M with respect to a multiplicative set S in R is described.

Goldie dimensions of quotient modules

AbstractFor an infinite cardinal ℵ an associative ring R is quotient ℵ<-dimensional if the generalized Goldie dimension of all right quotient modules of RR are strictly less than ℵ. This latter quotient property of RR is characterized in terms of certain essential submodules of cyclic modules being generated by less than ℵ elements, and also in terms of weak injectivity and tightness properties of certain subdirect products of injective modules. The above is the higher cardinal analogue of the known theory in the finite ℵ = ℵ0 case.

Pseudocomplemented Semilattices, Boolean Algebras, and Compatible Products

Pseudocomplemented semilattices are studied here from an algebraic point of view, stressing the pivotal role played by the pseudocomplements and the relationship between pseudocomplemented semilattices and Boolean algebras. Following the pattern of semiprime ring theory, a notion of Goldie dimension is introduced for complete pseudocomplemented lattices and calculated in terms of maximal uniform elements if they exist in abundance. Products in lattices with 0-element are studied and questions about the existence and uniqueness of compatible products in pseudocomplemented lattices, as well as about the abundance of prime elements in lattices with a compatible product, are discussed. Finally, a Yood decomposition theorem for topological rings is extended to complete pseudocomplemented lattices.

ENDOMORPHISM RINGS OF QUASI-PRINCIPALLY INJECTIVE MODULES

A right R-module M is called quasi-principally injective if every homomorphism from an M-cyclic submodule of M to M can be extended to M. Let M be a quasi-principally injective module which a self generator. In this paper we show that if such a module M has finite Goldie dimension, then S/J(S) is semisimple, where S = End(M R ). Furthermore if M is a self-generator quasi-principally injective module and M/soc(M) satisfies ACC on M-annihilator submodules, then J(S) is nilpotent. Some recent results obtained by Nicholson and Yousif are generalized.

Weakly Special Classes of Semiprime Rings

This paper investigates closure properties possessed by certain classes of finite subdirect products of prime rings. If ℳ is a special class of prime rings then the class ℳ∞ of all finite subdirect products of rings in ℳ is shown to be weakly special. A ring S is said to be a right tight extension [resp. tight extension] of a subring R if every nonzero right ideal [resp. right ideal and left ideal] of S meets R nontrivially. Every hereditary class of semiprime rings closed under tight extensions is weakly special. Each of the following conditions imposed on a semiprime ring yields a hereditary class closed under right tight extensions: ACC on right annihilators; finite right Goldie dimension; right Goldie. The class of all finite subdirect products of uniformly strongly prime rings is shown to be closed under tight extensions, answering a published question.