An extension of semicommutative rings via reflexivity

In this paper, we investigate locally finitely presented pure semisimple (hereditary) Grothendieck categories. We show that every locally finitely presented pure semisimple (resp., hereditary) Grothendieck category $\mathscr {A}$ is equivalent to the category of left modules over a left pure semisimple (resp., left hereditary) ring when $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, and every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. In fact, we show that there exists a bijection between Morita equivalence classes of left pure semisimple (resp., left hereditary) rings $\Lambda $ and equivalence classes of locally finitely presented pure semisimple (resp., hereditary) Grothendieck categories $\mathscr {A}$ that $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, and every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. To prove this result, we study left pure semisimple rings by using Auslander’s ideas. We show that there exists, up to equivalence, a bijection between the class of left pure semisimple rings and the class of rings with nice homological properties. These results extend the Auslander and Ringel–Tachikawa correspondence to the class of left pure semisimple rings. As a consequence, we give several equivalent statements to the pure semisimplicity conjecture.

In [1] we introduced the concept of a non-commutative local ring and studied the structure of such rings. Unfortunately, we were not able to show that the completion of a local ring was a semi-local ring. In this paper we propose to study a class of rings for which the above result is valid. This class of rings is the integral extensions -[4, 5]-of commutative local rings. This class of rings includes the important class of matrix rings over commutative local rings. In part 1 below we study some elementary properties of integral extensions and here we assume merely that the underlying ring is semi-local. In part 2 we discuss some questions of ideal theory for arbitrary local rings as well as for integral extensions. In a later paper we propose to utilize our results to study the deeper properties of these rings including a dimension theory for such rings. We are particularly indebted to the work of Nagata [6, 7, 8, 9] in the preparation of this paper.

Left localizable rings and their characterizations

1. Introduction. The concept of continuous modules owes its origin to Von Neumann's work on continuous geometries [16], and Utumi's work on continuous rings ([12], [13], [14]). L. Jeremy in [3] defined continuous modules, and others ([2], [5], [6], [7], [10]) later provided many interesting properties of such modules. A module is called continuous if (i) every submodule is essential in a summand and (ii) every submodule, isomorphic to a summand, is itself a summand. It can be easily seen, that while every quasi-injective module is continuous ([3], [5], [10]), the converse is not true. It is, therefore, natural to ask: For what classes of rings, is every continuous module quasi-injective? We provide here some classes of commutative rings, for which this is true. Our study is motivated by a general interest in identifying rings for which continuity is a non-trivial concept, and specifically, because only for such rings the existence of continuous hulls [7] poses a problem. We show that large classes of commutative rings, such as noetherian ones, and semiprimary ones with Jacobson radical square zero, have the property that every continuous module is quasi-injective. We give a necessary and sufficient condition, over an arbitrary commutative ring, for every uniform continuous module to be quasi-injective. We also provide necessary and sufficient conditions for some special classes of commutative rings, such as perfect rings, finitary rings, and chain rings, to have the property that every continuous module is quasi-injective. All our rings are commutative, have identity elements, and all modules are unitary rightmodules. Homomorphisms and endomorphisms act on the left. E(M)= EM denotes the injective hull of a module M. A c'M means that A is an essential submodule of M, a "summand" means a "direct summand", and e ~ stands for the annihilator in R of an element e of an R-module. A module A is said to be B-injective if every homomorphism from a submodule of B to A can be extended to B. We recall that a module M is called quasi-continuous (or 7r-injective) if M is closed under all idempotents of EndoRE(M) ([2], [3]). This holds if and only if every decomposition E(M)-- ~ @ E, induces a decomposition M= ~ @ (M hE,) tel tel ([2], Theorem 1.1). Every continuous module is quasi-continuous, and every quasi-continuous module satisfies the condition (i) of the definition of a continuous module. Let Mbe a module, with an injective hull E, then a continuous overmodule C of Mis called a continuous hull of M if Mc C ~ E, and if C c X holds for every continuous module M= X c E. We mention that this definition is independent of the choice of E [7].

In this paper, we study the semicommutative properties of rings with one-sided nilpotent structures. The concepts of left and right nilpotent semicommutative rings are defined and studied, which are a generalization of semicommutative rings. We show that the class of one-sided nilpotent semicommutative rings is strictly placed between the class of semicommutative rings and that of nil-semicommutative rings. As applications, we give some characterizations of semiprime rings from the point of view of left nilpotent semicommutative rings.

A necessary and sufficient condition on the endomorphism ring of a module for the module to have the finite exchange property is given. This condition is shown to be strictly weaker than a sufficient condition given by Warfield. The class of rings having these properties is equationally definable and is a natural generalization of the class of regular rings. Finally, it is observed that in the commutative case the category of such rings is equivalent with the category of ringed spaces ( X , R ) (X,\mathcal {R}) with X a Boolean space and R \mathcal {R} a sheaf of commutative (not necessarily Noetherian) local rings.

A necessary and sufficient condition on the endomorphism ring of a module for the module to have the finite exchange property is given. This condition is shown to be strictly weaker than a sufficient condition given by Warfield. The class of rings having these properties is equationally definable and is a natural generalization of the class of regular rings. Finally, it is observed that in the commutative case the category of such rings is equivalent with the category of ringed spaces $(X,\mathcal {R})$ with X a Boolean space and $\mathcal {R}$ a sheaf of commutative (not necessarily Noetherian) local rings.

This paper consists of three parts. The first is devoted to investigating the equivalence and left-right symmetry of several conditions known to characterize finite dimensional algebras which have a unique minimal faithful representation— QF- 3 3 algebras—in the class of left perfect rings. It is shown that the following conditions are equivalent and imply their right-hand analog: R R contains a faithful Σ \Sigma -injective left ideal, R R contains a faithful I I II -projective injective left ideal; the injective hulls of projective left R R -modules are projective, and the projective covers of injective left R R -modules are injective. Moreover, these rings are shown to be semi-primary and to include all left perfect rings with faithful injective left and right ideals. The second section is concerned with the endomorphism ring of a projective module over a hereditary or semihereditary ring. More specifically we consider the question of when such an endomorphism ring is hereditary or semihereditary. In the third section we establish the equivalence of a number of conditions similar to those considered in the first section for the class of hereditary rings and obtain a structure theorem for this class of hereditary rings. The rings considered are shown to be isomorphic to finite direct sums of complete blocked triangular matrix rings each over a division ring.

Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass’ conjecture is true for left duo rings. Call a ring R weakly left duo if for every r ε R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rrn(r) is two-sided. Recently, Xue [21] proved that Bass’ conjecture is still true for weakly left duo rings.

ABSTRACTLet p be a prime number. Let B be the idealization ℤ(+)ℤ∕pℤ and N: = pℤ(+)ℤ∕pℤ. Let B⊂R be a ramified (integral minimal) ring extension, with crucial maximal ideal 𝒩 and (necessarily) an element y such that R = B[y], y2∈B, y3∈B and y𝒩⊆𝒩. Then B is the only (commutative unital) ring properly contained between ℤ and R if and only if 𝒩 = N and either or py∉ℤ. Moreover, there are exactly three isomorphism classes of such rings R. Let A: = ℤ∕pαℤ for some integer α≥2. Let A⊂D⊂E be a tower of ramified ring extensions, (necessarily) with an element z such that E = D[z], z2∈D, z3∈D and zℳ⊆ℳ, where ℳ is the crucial maximal ideal of D⊂E. Then D is the only ring properly contained between A and E if and only if either z2∉A or pz∉A. There are at least two, but only finitely many, isomorphism classes of such rings E. We complete the characterization of the rings with exactly two proper (unital) subrings, including a classification up to isomorphism of these rings in the case of characteristic 0.

The main purposes of this paper are to investigate ℤ-injective rings with the representation extension property and its dual, to give a necessary and sufficient condition for a ℤ-injective ring to be an amalgamation base in the class of all rings and to determine structure of ℤ-injective Noetherian rings which are amalgamation bases. Further, in the class of all commutative rings, it is shown that a commutative ring has the representation extension property, if, and only if, it is an amalgamation base.

The class of semiprime left Goldie rings is a huge class of rings that contains many large subclasses of rings – semiprime left Noetherian rings, semiprime rings with Krull dimension, rings of differential operators on affine algebraic varieties and universal enveloping algebras of finite dimensional Lie algebras to name a few. In the paper, ‘Ring endomorphisms with large images,’ A. Leroy and J. Matczuk [Glasg. Math. J. 55 (2013), pp. 381–390] posed the following question: If a ring endomorphism of a semiprime left Noetherian ring has a large image, must it be injective? The aim of the paper is to give an affirmative answer to the Question of Leroy and Matczuk and to prove the following more general results. Theorem (Dichotomy). Each endomorphism of a semiprime left Goldie ring with large image is either a monomorphism or otherwise its kernel contains a regular element of the ring ( ⇔ \Leftrightarrow its kernel is an essential left ideal of the ring). In general, both cases are nonempty. Theorem. Every endomorphism with large image of a semiprime ring with Krull dimension is a monomorphism. Theorem (Affirmative answer to the Question of Leroy and Matczuk). Every endomorphism with large image of a semiprime left Noetherian ring is a monomorphism.

The Cartan determinant conjecture for left artinian rings was verified for quasihereditary rings showing detC(R) = detC(R/I), where I is a protective ideal generated by a primitive idempotent. This article identifies classes of rings generalizing the quasihereditary ones, first by relaxing the “projective” condition on heredity ideals. These rings, called left k-hereditary are all of finite global dimension. Next a class of rings is defined which includes left serial rings of finite global dimension, quasihereditary and left 1-hereditary rings, but also rings of infinite global dimension. For such rings, the Cartan determinant conjecture is true, as is its converse. This is shown by matrix reduction. Examples compare and contrast these rings with other known families and a recipe is given for constructing them.

We study classical right quotient rings and ordinary extensions of various kinds of 2-primal rings, constructing examples for situations that raise naturally in the process. We show: (1) Let R be a right Ore ring with P(R) left T-nilpotent. Then Q is a 2-primal local ring with P(Q)=J(Q) = {ab-1 ∈ Q | a ∈ P(R), b ∈ C(0)} if and only if C(0)=C(P(R))=R∖P(R), where Q is the classical right quotient ring of R. (2) Let R be a right Ore ring. Then R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R). As a consequence, if R is a right Noetherian ring, then R[[x]] is a domain whose classical right quotient ring is a division ring if and only if R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R).

The concept of central reflexive rings is introduced in this paper as a generalization of reflexive rings. Since every reflexive ring is central reflexive, we study the sufficient condition for a central reflexive ring to be a reflexive one. It is shown that every central reversible and hence every central symmetric ring is central reflexive, however converse implications are wrong. It is also proven that the polynomial ring R[x] is central reflexive if R is central reflexive and quasi-Armendariz. We have given some characterizations for an α-reflexive ring and also improved some of the results given by Zhao and Zhu [Extensions of α-reflexive rings, Asian-European J. Math.5(1) (2012) 1–10]. Finally we introduced the concept of central α-reflexive rings as a generalization of α-reflexive rings.