When every pure ideal is projective

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

This paper introduces the notion of essentially ADS (e-ADS) modules. Basic structural properties and examples of e-ADS modules are presented. In particular, it is proved that (1) The class of all e-ADS modules properly contains all ADS as well as automorphism invariant modules. e-ADS modules serves also as a tool for characterization of various classes of rings. It is shown that: (2) R R is a QF-ring if and only if every projective right R R -module is e-ADS; (3) R R is a semisimple Artinian ring if and only if every e-ADS module is injective. The final part of this paper describes properties of e-ADS rings, which allow to prove a criterion of e-ADS modules for non-singular rings: (4) Let R R be a right non-singular ring and Q Q be its the right maximal ring of quotients. Then R R is a right e-ADS ring if and only if either e Q ≇ ( 1 − e ) Q eQ\not \cong (1-e)Q for any idempotent e ∈ R e\in R or R ≅ M 2 ( A ) R\cong \mathbb {M}_2(A) for a suitable right automorphism invariant ring A A .

A left module M over a ring R is cofaithful in case there is an embedding of R into a finite product of copies of M. Our main result states that a semiprime ring R is left Goldie, that is, has a semisimple Artinian left quotient ring, if and only if R satisfies (i) every faithful left ideal is cofaithful and (ii) every nonzero left ideal contains a nonzero uniform left ideal. The proof is elementary and does not make use of the Goldie and LesieurCroisot theorems. We show that (i) and (ii) are Morita invariant. Moreover, (ii) is invariant under polynomial extensions, and so is (i) for commutative rings. Absolutely torsionfree rings are studied. The ring Q is a left classical quotient ring for the ring R C Q if every regular element (nondivisor of zero) of R is invertible in Q and if every element of Q is of the form b ~ιa where a,b E R and b is regular; in this case we also say that R is a left order in Q. A ring is said to be left Goldie if it has the ascending chain condition on left annihilators and has finite uniform dimension. (A left R -module has finite uniform dimension if it has no infinite direct sum of nonzero submodules, and it is said to be uniform if it is nonzero and any two nonzero submodules have a nontrivial intersection.) A theorem of Goldie [8,9] and Lesieur and Croisot [12] states that a ring is a left order in a semisimple Artinian ring if and only if it is semiprime and left Goldie. It is known that the ascending chain condition on left annihilators is not preserved under an equivalence of categories (Morita invariant); in fact, it does not go up to matrix rings. It is unknown whether being left Goldie is Morita invariant. In section two we give a proof of the theorem stated in the abstract, and in the prime case we give a proof which shows directly that such a ring is an order in a full matrix ring over a division ring. We also weaken the hypothesis of an important theorem on semiprime PI rings. In the third section we use these techniques to study absolutely torsion-free rings. In particular, we show that an absolutely torsionfree ring is Goldie if and only if it has a uniform left ideal, and that the endomorphism ring of a finitely generated projective module over an absolutely torsion-free ring is absolutely torsion-free.

The units of a ring R are defined by means of a multiplicative property, but in many cases they generate R additively. For example, it is shown in [5, Proposition 6] that if R is a semi-simple Artinian ring then every element of R is a sum of units if and only if the ring S = ℤ/2ℤ⊕ℤ/2ℤ is not a direct summand of R, where ℤ denotes the ring of integers. The theme of this paper is to investigate the corresponding situation concerning regular elements, i. e. elements which are not zero-divisors. We show that if R is a semi-prime right Goldie ring then every element of R is a sum of regular elements if and only if R does not have the ring S defined above as a direct summand (Corollary 2.9). We also characterise those Noetherian rings R such that every element of R is a sum of regular elements (Theorem 2. 6). The characterisation is in terms of the nature of certain prime factor rings of R, and it is again the presence of the ring S, this time in a particular way as a factor ring of R, which prevents R from being generated by its regular elements. If R has no non-zero Artinian one-sided ideals or if 2 is a regular element of R, then every element of R is a sum of regular elements (Corollaries 2. 5 and 2. 7). As an application we show in Section 3 that, for many Noetherian rings R, the set of elements of R which are divisible by every regular element of R is a two-sided ideal of R.

Following O'Meara's result [Journal of Algebra and Its Applications, 13 (2014) 8], it follows that the block matrix A = B 0 0 0 M n+r (R), B M n (R), r 1, over a von Neumann regular separative ring R, is a product of idempotent matrices.Furthermore, this decomposition into idempotents of A also holds when B is an invertible matrix and R is a GE ring (defined by Cohn [New mathematical monographs: 3, Cambridge University Press, 2006]).As a consequence, it follows that if there exists an example of a von Neumann regular ring R over which the matrix A = B 0 0 0 M n+r (R), where B M n (R), r 1, cannot be expressed as a product of idempotents, then R is not separative, thus providing an answer to an open question whether there exists a von Neumann regular ring which is not separative.The paper concludes with a counter example related to an open question whether every totally nonnegative matrix is a product of nonnegative idempotent matrices.

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.

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no nonzero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so-called e-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the [Formula: see text]-injective semimodule, and the [Formula: see text]-injective semimodules through several implications, examples and counter examples. Moreover, we show that every semimodule over an arbitrary semiring can be embedded in a [Formula: see text]-[Formula: see text]-injective semimodule.

Almost slender rings and compact rings

Given a finite (associative, unital) ring R, let K(R) denote the set of polynomials in R[x] that send each element of R to 0 under evaluation. We study K(R) and its elements. We conjecture that K(R) is a two-sided ideal of R[x] for any finite ring R, and prove the conjecture for several classes of finite rings (including commutative rings, semisimple rings, local rings, and all finite rings of odd order). We also examine a connection to sets of integer-valued polynomials.

In part I we introduce vertex rings , which bear the same relation to vertex algebras (or VOAs) as commutative, associative rings do to commutative, associative algebras over C \mathbf {C} . We show that vertex rings are characterized by Goddard axioms . These include a generalization of the translation-covariance axiom of VOA theory that involves a canonical Hasse-Schmidt derivation naturally associated to any vertex ring. We give several illustrative applications of these axioms, including the construction of vertex rings associated with the Virasoro algebra. We consider some categories of vertex rings, and the rôle played by the center of a vertex ring. In part II we extend the theory of Pierce bundles associated to a commutative ring to the setting of vertex rings. This amounts to the construction of certain reduced étale bundles of vertex rings functorially associated to a vertex ring. We introduce von Neumann regular vertex rings as a generalization of von Neumann regular commutative rings; we obtain a characterization of this class of vertex rings as those whose Pierce bundles are bundles of simple vertex rings.

Publisher Summary This chapter discusses the semi-regular, weakly regular and π-regular rings. All rings are associative and have nonzero identity elements. Every strongly π-regular ring is π-regular. The factor ring of the integers with respect to the ideal generated by the integer four is a strongly re-regular ring which is not a regular ring. For a ring A, the Jacobson radical of A is denoted by J (A) which is equal to the intersection of all maximal right ideals of A by definition. A ring A is said to be semi-regular if A / J (A) is a regular ring, and all idempotents of A / J (A) are images of idempotents of A for the natural epimorphism A → A / J (A). The classes of semi-regular and π-regular rings are quite large. Every regular ring is a semi-regular π-regular ring. All right or left Artinian rings are semi-regular π-regular rings, and the endomorphism ring of any injective module is a semi-regular ring.

We introduce and study lattice-finite Noetherian rings and show that they form a onedimensional analogue of representation-finite Artinian rings. We prove that every lattice-finite Noetherian ring R has Krull dimension ≼ 1, and that R modulo its Artinian radical is an order in a semi-simple ring. Our main result states that maximal overorders of R exist and have to be Asano orders, while they need not be fully bounded. This will be achieved by means of an idempotent ideal I(R), an invariant or R which is new even for classical orders R. This ideal satisfies I(R) = R whenever R is maximal.

A right [Formula: see text]-module [Formula: see text] is called to satisfy condition [Formula: see text] if, for every [Formula: see text] and [Formula: see text], there exists [Formula: see text] such that [Formula: see text] and if [Formula: see text] is a direct summand of [Formula: see text], then [Formula: see text] is a direct summand of [Formula: see text]. In this paper, we give some properties of rings and modules to satisfy condition [Formula: see text]. Moreover, their connections with von Neumann regular rings, Hereditary rings, Noetherian rings and (semi)artinian rings are addressed.

We prove that if the direct sum of a family of semimodules over a semiring S is an injective semimodule or if the direct product of a family of semimodules over S is a projective semimodule, then the cardinality of the subfamily consisting of all semimodules which are not modules is strictly less than the cardinality of S. As a consequence, we obtain semiring analogs of well-known characterizations of classical semisimple, quasi-Frobenius, and one-sided Noetherian 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.