On FC-Purity and I-Purity of Modules and Köthe Rings

In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Köthe ring. Finally, we introduce FC-pure flatness and I-pure flatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module.     

Rings whose cyclic modules are lifting and ⊕-supplemented

It is proved that a semiperfect module is lifting if and only if it has a projective cover preserving direct summands. Three corollaries are obtained: (1) every cyclic module over a ring R is lifting if and only if every cyclic R-module has a projective cover preserving direct summands; (2) a ring R is artinian serial with Jacobson radical square-zero if and only if every (2-generated) R-module has a projective cover preserving direct summands; (3) a ring R is a right (semi-)perfect ring if and only if (cyclic) lifting R-module has a projective cover preserving direct summands, if and only if every (cyclic) R-module having a projective cover preserving direct summands is lifting. It is also proved that every cyclic module over a ring R is ⊕-supplemented if and only if every cyclic R-module is a direct sum of local modules. Consequently, a ring R is artinian serial if and only if every left and right R-module is a direct sum of local modules.     

On semilocal modules and rings

It is well-known that a ring Ris semiperfect if and only if _RR (orR_R ) is a supplemented module. Considering weak supplementsinstead of supplements we show that weakly supplemented modules Mare semilocal (i.e.M/Rad(M) is semisimple) and that R is a semilocal ring if and only if _RR (orR_R ) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modules are weakly supplemented if and only if they have finite hollow dimension (or are semilocal).     

C-Pure Projective Modules

This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ring R is left Noetherian if and only if every C-pure projective left R-module is pure projective. Also, over a left hereditary Noetherian ring R, a left R-module M is C-pure projective if and only if M = N ⊕ P, where N is a direct sum of cyclic modules and P is a projective left R-module. The relationship C-purity with purity and RD-purity are also studied. It is shown that if R is a local duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is a principal ideal ring. If R is a left perfect duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is left Köthe (i.e., every left R-module is a direct sum of cyclic modules). Also, it is shown that for a ring R, if every C-pure projective left R-module is RD-projective, then R is left Noetherian, every p-injective left R-module is injective and every p-flat right R-module is flat. Finally, it is shown that if R is a left p.p-ring and every C-pure projective left R-module is RD-projective, then R is left Noetherian hereditary. The converse is also true when R is commutative, but it does not hold in the noncommutative case.     

Integrally Closed Domains with Only Finitely Many Star Operations

We prove that an integrally closed domain R admits only finitely many star operations if and only if R satisfies each of the following conditions: (1) R is a Prüfer domain with finite character, (2) all but finitely many maximal ideals of R are divisorial, (3) only finitely many maximal ideals of R contain a nonzero prime ideal that is contained in some other maximal ideal of R, and (4) if P ≠ (0) is the largest prime ideal contained in a (necessarily finite) collection of maximal ideals of R, then the prime spectrum of R/P is finite.     

Partial Skew Polynomial Rings and Goldie Rings

We prove that if R is a semiprime ring and α is a partial action of an infinite cyclic group on R, then R is right Goldie if and only if R[x; α] is right Goldie if and only if R?x; α? is right Goldie, where R[x; α] (R?x; α?) denotes the partial skew (Laurent) polynomial ring over R. In addition, R?x; α? is semiprime while R[x; α] is not necessarily semiprime.     

On rings in which every maximal one-sided ideal contains a maximal ideal

Given a ring R, consider the condition: (^*) every maximal right ideal of R contains a maximal ideal of R. We show that, for a ring R and 0 ≠ e ² = e ∈ R such that ele ? eRe every proper ideal I of R R satisfies (^*) if and only if eRe satisfies (^*). Hence with the help of some other results, (^*) is a Morita invariant property. For a simple ring R R[x] satisfies (^*) if and only if R[x] is not right primitive. By this result, if R is a division ring and R[x] satisfies (^*), then the Jacobson conjecture holds. We also show that for a finite centralizing extension S of a ring R R satisfies (^*) if and only if S satisfies (^*).     

Additive Set of Idempotents in Rings

Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, f ∈ I(R) (e ≠ f), e + f ∈ I(R), and M(R) is additive in I(R) if for all e, f ∈ M(R)(e ≠ f), e + f ∈ I(R). In this article, the following points are shown: (1) I(R) is additive if and only if I(R) is multiplicative and the characteristic of R is 2; M(R) is additive in I(R) if and only if M(R) is orthogonal. If 0 ≠ ef ∈ I(R) for some e ∈ M(R) and f ∈ I(R), then ef ∈ M(R), (2) If R has a complete set of primitive idempotents, then R is a finite product of connected rings if and only if I(R) is multiplicative if and only if M(R) is additive in I(R).     

Gelfand Factor Rings and Weak Zariski Topologies

Let R be any ring with identity. Let N(R) (resp. J(R)) denote the prime radical (resp. Jacobson radical) of R, and let Spec _r (R) (resp. Spec _l (R), Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all left prime ideals, all maximal right ideals, all right primitive ideals) of R. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec _r (R) (with weak Zariski topology). The following results are obtained: (1) R/N(R) is a Gelfand ring if and only if Spec _r (R) is a normal space if and only if Spec _l (R) is a normal space; (2) R/J(R) is a Gelfand ring if and only if every right prime ideal containing J(R) is contained in a unique maximal right ideal.     

PRINCIPAL IDEAL DOMAINS AND EUCLIDEAN DOMAINS HAVING 1 AS THE ONLY UNIT

We consider a question raised by Mowaffaq Hajja about the structure of a principal ideal domain R having the property that 1 is the only unit of R. We also examine this unit condition for the case where R is a Euclidean domain. We prove that a finitely generated Euclidean domain having 1 as its only unit is isomorphic to the field with two elements F ₂ or to the polynomial ring F ₂[X]. On the other hand, we establish existence of finitely generated principal ideal domains R such that 1 is the only unit of R and R is not isomorphic to F ₂ or to F ₂[X]. We also construct principal ideal domains R of infinite transcendence degree over F ₂ with the property that 1 is the only unit of R.

     

Krull Dimension in Power Series Ring Over an Almost Pseudo-Valuation Domain

Let R be an integral domain. We say that R is a star-domain if R has at least a height one prime ideal and if for each height one prime ideal P of R, R satisfies the acc on P-principal ideals (i.e., ideals of the form aP, a ∈ R). We prove that if R is an APVD with nonzero finite Krull dimension, then the power series ring R[[X]] has finite Krull dimension if and only if R is a residually star-domain (i.e., for each nonmaximal prime ideal P of R, R/P is a star-domain) if and only if R[[X]] is catenarian.     

Submodules of Direct Sums of Compressible Modules

Let R be a ring. A right R-module M is called “essentially compressible” if it embeds in each of its essential submodules. Also a module X _R is called “completely essentially compressible” if every submodule of X _R is an essentially compressible R-module. In this aricle, it is shown that a right R-module M embeds in a direct sum of compressible right R-modules if and only if M _R is essentially compressible and every nonzero essentially compressible submodule of M _R contains a compressible submodule. Every essentially compressible R-module is shown to be retractable. Moreover, if either R _R has Krull dimension, or R is Morita equivalent to a right duo ring, then a right R-module embeds in a direct sum of compressible right R-modules if and only if it is completely essentially compressible.     

Semilocal rings with <Emphasis Type="Italic">n</Emphasis>-Engel multiplicative group

An associative ring R with unit element is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that the multiplicative group R* of a semilocal ring R generated by R* satisfies an n-Engel condition for some positive integer n if and only if R is m-Engel as a Lie ring for some positive integer m depending only on n.Received: 21 January 2003     

On commutative rings which are strongly prüfer

Abstract

As defined by Nicholson [Nicholson, W. K. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229:269–278] an element of a ring R is clean if it is the sum of a unit and an idempotent, and a subset A of R is clean if every element of A is clean. It is shown that a semiprimitive Gelfand ring R is clean if and only if Max(R) is zero-dimensional; if and only if for each M ∈ Max(R), the intersection all prime ideals contained in M is generated by a set of idempotents. We also give several equivalent conditions for clean functional rings. In fact, a functional ring R is clean if and only if the set of clean elements is closed under sum; if and only if every zero-divisor is clean; if and only if; R has a clean prime ideal.     

On Finitely Annihilated Modules and Artinian Rings

Let R be a ring. An R-module M is finitely annihilated if the annihilator of M is the annihilator of a finite subset of M. It is proved that if R has right socle S then the ring R/S is right Artinian if and only if every singular right R-module is finitely annihilated. Moreover, a right Noetherian ring R is right Artinian if and only if every uniform right R-module is finitely annihilated. In addition, a (right and left) Noetherian ring is (right and left) Artinian if and only if every injective right R-module is finitely annihilated. This paper will form part of the Ph.D. thesis at the University of Glasgow of the second author. He would like to thank the EPSRC for their financial support     

Left APP differential polynomial rings

A ring R is called a left APP-ring if for each element a∈R, the left annihilator l_R(Ra) is right s-unital as an ideal of R or equivalently R∕l_R(Ra) is flat as a left R-module. In this paper, we show that for a ring R and derivation δ of R, R is left APP if and only if R is δ-weakly rigid and the differential polynomial ring R[x;δ] is left APP. As a consequence, we see that if R is a left APP-ring, then the n^th Weyl algebra over R is left APP. Also we define δ-left APP (resp. p.q.-Baer) rings and we show that R is left APP (resp. p.q.-Baer) if and only if for each derivation δ of R, R is δ-weakly rigid and δ-left APP (resp. p.q.-Baer). Finally we prove that R[x;δ] is left APP (resp. p.q.-Baer) if and only if R is δ-left APP (resp. p.q.-Baer).     

On a Generalization of Injective Rings

Abstract

A ring R is called left IP-injective if every homomorphism from a left ideal of R into R with principal image is given by right multiplication by an element of R. It is shown that R is left IP-injective if and only if R is left P-injective and left GIN (i.e., r(I ∩ K) = r(I) + r(K) for each pair of left ideals I and K of R with I principal). We prove that R is QF if and only if R is right noetherian and left IP-injective if and only if R is left perfect, left GIN and right simple-injective. We also show that, for a right CF left GIN-ring R, R is QF if and only if Soc(R _R ) ? Soc( _R R). Two examples are given to show that an IP-injective ring need not be self-injective and a right IP-injective ring is not necessarily left IP-injective respectively.     

Supplement Submodules of Injective Modules

An R-module M is called almost injective if M is a supplement submodule of every module which contains M. The module M is called F-almost injective if every factor module of M is almost injective. It is shown that a ring R is a right H-ring if and only if R is right perfect and every almost injective module is injective. We prove that a ring R is semisimple if and only if the R-module R _R is F-almost injective.     

Ring elements as sums of units

In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph Γ(R) is a power of 2.