Rings whose cyclic modules have restricted injectivity domains

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

In the first two sections of this chapter, we focus our attention on two special classes of rings, namely, local rings and semilocal rings. By definition, a ring R is local if R/rad R is a division ring, and R is semilocal if R/rad R is a semisimple ring. Thus, local rings include all division rings, and semilocal rings include all left or right artinian rings. The basic properties of local and semilocal rings are developed, respectively, in §19 and §20. We shall see, for instance, that local rings are connected with the problem of the uniqueness of Krull-Schmidt decompositions, and that semilocal rings are connected with the problem of “cancellation” of modules.

A ring R is called right PCI if every proper cyclic right R-module is injective, i.e. if C is a cyclic right R-module then CR ≅ RR or CR is injective. By [2] and [3], if R is a non-artinian right PCI ring then R is a right hereditary right noetherian simple domain. Such a domain is called a right PCI domain. The existence of right PCI domains is guaranteed by an example given in [2]. As generalizations of right PCI rings, several classes of rings have been introduced and investigated, for example right CDPI rings, right CPOI rings (see [8], [6]). In Section 2 we define right PCS, right CPOS and right CPS rings and study the relationship between all these rings.

Local rings whose modules are almost serial

Perfect modules over cohen-macaulay local rings

Two obvious classes of quasi-injective modules are those of semisimples and injectives. In this paper, we study rings with no quasi-injective modules other than semisimples and injectives. We prove that such rings fall into three classes of rings, namely, (i) QI-rings, (ii) rings with no middle class, or (iii) rings that decompose into a direct product of a semisimple Artinian ring and a strongly prime ring. Thus, we restrict our attention to only strongly prime rings and consider hereditary Noetherian prime rings to shed some light on this mysterious case. In particular, we prove that among these rings, QIS-rings which are not of type (i) or (ii) above are precisely those hereditary Noetherian prime rings which are idealizer rings from non-simple QI-overrings.

Chapter 7 continues to develop the theory of rings and studies chain conditions for ideals of a ring. The motivation came from an interesting property of the ring of integers Z that its every ascending chain of ideals terminates. This interesting property of Z was first recognized by the German mathematician Emmy Noether (1882–1935). This property leads to the concept of Noetherian rings. On the other hand, Emil Artin (1898–1962) showed that there are some rings in which every infinite descending chain of ideals terminates. Such rings are called Artinian rings. The ring Z has an infinite descending chain of ideals. This property of Z shows that Z is not an Artinian ring. In this chapter special classes of rings are also studied and deeper results on ideal theory are proved. Some finiteness condition for ideals is imposed and Noetherian rings are introduced which are versatile. The most convenient equivalent formulation of the Noetherian requirement is that the ideals of the ring satisfy the ascending chain condition. A connection between a Noetherian domain and a factorization domain is established. Moreover, the Hilbert Basis Theorem is proved. This theorem has made a revolution and has given an extensive class of Noetherian rings. Its application to algebraic geometry is also discussed. The study in this chapter culminates in rings with a descending chain condition for ideals, which determines their ideal structure.

On Auslander's depth formula

Applications of dualizing complexes to Buchsbaum rings

Almost slender rings and compact rings

Every indecomposable module over a generalized uniserial ring is uniserial, hence local. This motivates one to study rings R satisfying the condition (*): R is a right artinian ring such that every finitely generated, indecomposable right R-module is local. The rings R satisfying (*) have been recently studied by Singh and Al-Bleahed (2004), they have proved some results giving the structure of local right R-modules. In this paper some more structure theorems for local right R-modules are proved. Examples given in this paper show that a rich class of rings satisfying condition (*) can be constructed. Using these results, it is proved that any ring R satisfying (*) is such that mod-R is of finite representation type. It follows from a theorem by Ringel and Tachikawa that any right R-module is a direct sum of local modules. If M is right module over a right artinian ring such that any finitely generated submodule of any homomorphic image of M is a direct sum of local modules, it is proved that it is a direct sum of local modules. This provides an alternative proof for that any right module over a right artinian ring R satisfying (*) is a direct sum of local modules.

Throughout this paper the word “ring” will mean an associative ring which need not have an identity element. There are Artinian rings which are not Noetherian, for example C(p∞) with zero multiplication. These are the only such rings in that an Artinian ring R is Noetherian if and only if R contains no subgroups of type C(p∞) [1, p. 285]. However, a certain class of Artinian rings is Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with an identity element is Noetherian [3, p. 69]. The proofs of these theorems involve the method of “factoring through the nilpotent Jacobson radical of the ring”. In this paper we state necessary and sufficient conditions for an Artinian ring (and an Artinian module) to be Noetherian. Our proof avoids the concept of the Jacobson radical and depends primarily upon the concept of the length of a composition series. As a corollary we obtain the result of Hopkins.

The polynomial ring over a Goldie ring need not be a Goldie ring

In this paper, we study endo-reduced modules as modules whose endomorphism rings have no nonzero nilpotent elements. We characterize their properties for different classes of modules, including [Formula: see text]-non-singular modules, multiplication modules and finitely generated modules over commutative Dedekind domains. In the subcategory of finitely generated modules, it is shown that the class of rings [Formula: see text] for which every faithful multiplication [Formula: see text]-module is endo-reduced is precisely that of reduced rings; while the class of rings [Formula: see text] for which every multiplication [Formula: see text]-module is endo-reduced is precisely that of von Neumann regular rings. Characterizations of when an endo-reduced module will be a reduced module are given. We prove that a finitely generated module over a principal ideal domain (PID) is endo-reduced exactly if it is either a semisimple module with pair-wise non-isomorphic submodules or a torsion-free module which is isomorphic to the underlying ring.

It is well known that Hubert's function of a homogeneous ideal in the ring of polynomials K[x0, …, xm], where K is a field and x0, …, xm are independent indeterminates over K, is, for large values of r, a polynomial in r of degree equal to the projective dimension of (1). Samuel (4) and Northcott (2) have both shown that if the field K is replaced by an Artin ring A, is still a polynomial in r for large values of r. Applying this generalization Samuel (4) has shown that in a local ring Q the length of an ideal qρ, where q is a primary ideal belonging to the maximal ideal m of Q, is, for sufficiently large values of ρ, a polynomial in ρ whose degree is equal to the dimension of Q.