Cyclic R-module Research Articles

F-semi-perfect modules

Let R be a ring with Jacobson radical J. Bass [ 1 ] called R semi-perfect if the factor ring R/J is semi-simple Artinian and every idempotent of R/J can be lifted to an idempotent of R. He showed that R is semi-perfect if and only if every cyclic left R-module has a projective cover and also that this so (if and) only if every finitely generated left R-module has a projective cover. The concept of semi-perfect rings has since then been generalized in two directions: to semi-perfect modules by Mares [3] and to F-semiperfect rings by Oberst and Schneider [6]. A left R-module is called a semi-perfect module if it is projective and every homomorphic image of it has a projective cover. Thus trivially R is a semi-perfect ring if and only if R is semi-perfect as a left R-module, and every finitely generated projective left module over a semi-perfect ring is semi-perfect. The following characterization was obtained: a projective left R-module P is semi-perfect if and only if JP is small in P, the factor module P/JP is completely reducible (=semi-simple), and every direct decomposition of P/JP can be lifted to that of P. On the other hand, a ring R is defined to be F-semi-perfect if the factor ring R/J is a regular ring (in the sence of von Neumann) and every idempotent of R/J can be lifted to an idempotent of R. It was proved that the F-semi-perfectness of R is equivalent to either of the following conditions: (a) every factor module R/Ra has a projective cover for a in R, or (b) every finitely presented left R-module has a projective cover. We attempt in this paper to generalize the concept of F-semi-perfect rings to modules along the line from semi-perfect rings to semi-perfect modules. Namely we call a left R-module P an F-semi-perfect module if B is projective and if, for every endomorphism s of P, the factor module P/Ps

A characterization of rings with Krull dimension

Throughout this paper rings will mean associative rings with identity and all modules are assumed to be unitary. As is well known, cyclic modules play an important role in ring theory. Many nice properties of rings can be characterized by their cyclic modules, see for example [ 1, 2, 9-111. One of the most important results in this direction is the result of Osofsky [S, Theorem] which says that a ring R is semisimple (i.e., R is right Artinian with zero Jacobson radical) if and only if every cyclic right R-module is injective. Starting from this and in connection with a result of Vamos [ 1 l] the following result has been recently obtained in [2, Theorem 1.11: A ring R is right Artinian if and only if every cyclic right R-module is a direct sum of an injective module and a finitely cogenerated module. In the present paper we follow this investigation and aim to show a similar result for rings with Krull dimension. We consider the following condition about a ring R:

A generalization of right pci rings

By a well-known result of Osofsky [6, Theorem] a ring R is semisimple (i.e. R is right artinian and the Jacobson radical of R is zero) if and only if every cyclic right R-module is injective. Starting from this, a larger class of rings has been introduced and investigated, namely the class of right PCI rings. A ring R is called right PCI if every proper cyclic right R- module is injective (proper here means not being isomorphic to RR). By [l] and [Z], a right PCI ring is either semisimple or it is a right noetherian, right hereditary simple ring. The latter ring is usually called a right PCI domain. In this paper we consider the similar question in studying rings whose cyclic right modules satisfy some decomposition property. The starting point is a theorem recently proved in 13, Theorem 1.1): A ring R is right artinian if and only if every cyclic right R- module is a direct sum of an injective module and a finitely cogenerated module.

A Decomposition of Rings Generated by Faithful Cyclic Modules

AbstractA ring R is said to be generated by faithful right cyclics (right finitely pseudo-Frobenius), denoted by GFC (FPF), if every faithful cyclic (finitely generated) right R-module generates the category of right R-modules. The class of right GFC rings includes right FPF rings, commutative rings (thus every ring has a GFC subring - its center), strongly regular rings, and continuous regular rings of bounded index. Our main results are: (1) a decomposition of a semi-prime quasi-Baer right GFC ring (e.g., a semiprime right FPF ring) is achieved by considering the set of nilpotent elements and the centrality of idempotnents; (2) a generalization of S. Page's decomposition theorem for a right FPF ring.

Rings that are FGC relative to filters of ideals

All our rings will be commutative with identity not equal to zero. Also R will always denote a ring. is a filter of ideals of R if is a nonempty set of ideals of R satisfying: I∈ and J is an ideal of R with I⊂J, then J∈, and if I, J∈ then I∩J∈. A Gabriel topology of R is a filter of ideals of R satisfying: if J∈ and I is an ideal of R with (I:x)∈ for all x∈J, then I∈. See the B. Stenström text [6]. We say that a ring R is an FGC ring if every finitely generated R-module is a direct sum of cyclic R-modules. Use mspec R for the set of all maximal ideals of R.

Some remarks on multiplication modules

In this note all rings are commutative rings with identity and all modules are unital. Let R be a commutative ring with identity. An R-module M is called a multiplication module provided for each submodule N of M there exists an ideal I of R such that N = I M. Various properties of multiplication modules are considered. If there is a common theme it is that the methods used generalise results of Naoum and Hasan proved using matrix methods. 1. Sums and intersections. Let R be a commutative ring with identity and M a unital R-module. The annihilator of M is denoted ann (M) and for any m ~ M the annihilator ofm is denoted ann(m). IfN is a submodule of M then (N: M) denotes the ideal ann(M/N) of R, that is (N :M) = {r e R: rM c= N}. An R-module M is called a multiplication module provided for each submodule N of M there exists an ideal I of R such that N = I M. It is clear that every cyclic R-module is a multiplication module. Let N be a submodule of a multiplication module M. There exists an ideal I of R such that N = I M. Note that I~(N:M) and N=IM=(N:M) M~N so that N=(N:M) M. It follows that an R-module M is a multiplication module if and only if N = (N : M) M for all submodules N of M. An ideal A of R which is a multiplication module is called a multiplication ideal. Let P be a maximal ideal of a ring R. An R-module M is called P-torsion provided for each m ~ M there exists p E P such that (1 - p) m = 0. On the other hand M is called P-cyclic provided there exist x ~ M and q E P such that (1 - q) M ~ R x. Our starting point is the following result taken from [3, Theorem 1.2].

Three homotopy theories for cyclic modules

1.1. The main result. Let R be a ring with 1 +O and let C denote the category of the cyclic R-modules of Connes, i.e., [3, 61 simplicial R-modules X with, in each dimension n LO, an extra degeneracy map s,+, : X,+X,,+, satisfying the usual identities, except that the identities des,+, = ,, ,, s d : X,-X,, (n LO) are replaced by (dos,+,)“+‘= id :X,-X, (n10). Such a cyclic R-module has not only its underlying simplicial R-module, which we will denote by j *X, but also an underlying cosimplicial R-module k*X (with di=s,_i: X,,_i+Xn and s’=d,_i: Xn+XnPi). As a result, there are three obvious homotopy theories which one can associate with C; these correspond to three possible criteria for calling a map f: X-+X’ in C a ‘weak equivalence’ [6, Q 71. First there is the one-sided homotopy theory in which the weak equivalences are the maps f which induce isomorphisms xi j *X= n, j *X (ilO) on the homotopy groups of the underlying simplicial modules. Next, there is a dual one-sided theory in which the weak equivalences are the maps f which induce isomorphisms ?k*Xn’k*X’ (ir0) on the cohomotopy groups of the underlying cosimplicial modules. Finally, there is a strong or two-sided theory in which the weak equivalences are the maps f which induce isomorphisms on both the homotopy groups of the underlying simplicial modules and the cohomotopy groups of the underlying cosimplicial modules. The main aim of this paper is to show that each of these three homotopy theories is equivalent to a corresponding homotopy theory of differential graded modules over a graded exterior R-algebra. This implies that from any of the above three points of view the study of cyclic R-modules is equivalent to the study of the classical homological algebra of certain chain complexes. In more detail:

Rings over which all cyclic modules are poorly injective

The fundamental result of the paper is Theorem I, asserting that over a right Noetherian ring R, the poor injectivity of all cyclic modules is equivalent to the fact that R is a direct sum of a semisimple Artinian ring and a finite number of uniserial rings and hereditary Noetherian left-invariant domains. (We note that the injectivity of all cyclic R-modules over an arbitrary ring R is equivalent to the classical semisimplicity of the ring R [2].)

Q-Hypercyclic Rings

A ring R is called q-hypercyclic (hypercyclic) if each cyclic ring R-module has a cyclic quasi-injective (injective) hull. A ring R is called a qc-ring if each cyclic right R-module is quasi-injective. Hypercyclic rings have been studied by Caldwell [4], and by Osofsky [12]. A characterization of qc-rings has been given by Koehler [10]. The object of this paper is to study q-hypercyclic rings. For a commutative ring R, R can be shown to be q-hypercyclic (= qc-ring) if R is hypercyclic. (Theorems 4.2 and 4.3). Whether a hypercyclic ring (not necessarily commutative) is q-hypercyclic is considered in Theorem 3.11 by showing that a local hypercyclic ring R is q-hypercyclic if and only if the Jacobson radical of R is nil. However, we do not know if there exists a local hypercyclic ring with nonnil radical [12]. Example 3.10 shows that a q-hypercyclic ring need not be hypercyclic.

Rings Characterized by their Cyclic Modules

A ring R (with identity element) is called a right PCI-ring if and only if every proper cyclic right R-module is injective; that is, if C is a cyclic right R-module then either C ≌ R or C is injective. Faith [3, Theorems 14 and 17] (or see [2, Proposition 6.12 and Theorem 6.17]) proved that if a ring R is a right PCI-ring then R is semiprime Artinian or R is a simple right semihereditary right Ore domain. These latter rings we shall call simple rightPCI-domains. Examples of non-Artinian simple right PCI-domains were produced by Cozzens [1]. The object of this paper is to examine rings with similar properties and thus extend Faith's results.

On rings with quasi-injective cyclic modules

A ring R is called a qc-ring if each cyclic R-module is quasi-injective. For various properties of these rings we refer to Ahsan (1) and Koehler (15). In this paper we shall obtain some additional results related to qc-rings. The scheme of the paper is as follows. Section 2 contains various preliminary definitions and results. In Section 3, we shall prove that every commutative hypercyclic ring is a qc-ring. In this section, we shall also show that a qc-ring which satisfies the ascending chain condition on its annihilators has nilpotent Jacobson-radical. Finally, in Section 4, we shall study rings all of whose proper factor rings are qc. Such rings will be called “ restricted qc ”.

A note on modules over a commutative regular ring

An example is given of a commutative, von Neumann regular ring R, which has a module A satisfying the following conditions: (1) T ( A ) = { a ∈ A | ( 0 : a ) T(A) = \{ a \in A|(0:a) is an essential ideal of R} is a cyclic R-module; (2) A / T ( A ) A/T(A) is a cyclic R-module; and (3) T ( A ) T(A) is not a direct summand of A. This answers in the negative a question raised by R. S. Pierce.