Baer, Quasi-Baer Modules, and Their Applications

The well-known notion of an extending module is closely linked to that of a Baer module. A right [Formula: see text]-module [Formula: see text] is called extending if every submodule of [Formula: see text] is essential in a direct summand. On the other hand, a right [Formula: see text]-module [Formula: see text] is called Baer if for all [Formula: see text], [Formula: see text] where [Formula: see text]. In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc. 21(2) (1980) 434–444.] in terms of modules and showed the connections between Baer and extending modules via the result: “a module[Formula: see text] is[Formula: see text]-nonsingular extending if and only if[Formula: see text] is[Formula: see text]-cononsingular Baer”. [Formula: see text] is called [Formula: see text]-nonsingular if [Formula: see text] such that [Formula: see text], [Formula: see text]. Moreover, [Formula: see text] is called [Formula: see text]-cononsingular if for any [Formula: see text] with [Formula: see text] for all [Formula: see text], implies [Formula: see text]. In view of this result, every Baer module which happens to be [Formula: see text]-cononsingular will automatically become an extending module. In this paper, our main focus is the study of [Formula: see text]-cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of [Formula: see text]-cononsingularity while sufficient knowledge exists about the other three remaining notions in the preceding result. Moreover, we introduce the notion of special extending (or sp-extending, for short) of a module and show that the class of [Formula: see text]-cononsingular modules properly contains the class of extending modules and the class of special extending modules. Among other results, we obtain a new analogous version for the Rizvi–Roman’s result which illustrates the close connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided.

Whitehead modules

For an endomorphism of R, in (1), a module MR is called -compatible if, for any m 2 M and a 2 R, ma = 0 i m (a) = 0, which are a generalization of -reduced modules. We study on the relationship between the quasi-Baerness and p.q.- Baer property of a module MR and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of (2) and some results in (9). In particular, we show: for an -compatible module MR (1) MR is p.q.-Baer module i M(x; )R(x; ) is p.q.-Baer module. (2) for an automorphism of R, MR is p.q.-Baer module i M(x,x 1 ; )R(x,x 1; ) is p.q.- Baer module.

In this paper we introduce dual [Formula: see text]-Baer modules and give a characterization of them. Let [Formula: see text] be a module and [Formula: see text] a fully invariant submodule of [Formula: see text]. [Formula: see text] is called dual [Formula: see text]-Baer if for every family of endomorphisms [Formula: see text] of [Formula: see text], [Formula: see text] is a direct summand of [Formula: see text]. We prove that [Formula: see text] is dual [Formula: see text]-Baer if and only if [Formula: see text] for some submodule [Formula: see text] of [Formula: see text] with [Formula: see text] dual Baer. We obtain a positive solution for the Schröder–Bernstein problem for certain dual [Formula: see text]-Baer modules.

The question of whether all projective modules (or all finitely generated projective modules) over a given ring are free or extended is solved positively only for relatively small class of rings. In 1976, a significant progress was achieved in this area of research: D. Quillen and A. A. Suslin simultaneously and independently of one another proved that all projective modules over algebras of polynomials over fields are free modules, which was a solution to the well-known J.-P. Serre’s problem. Later similar results were obtained by various researchers for projectivemodules over ringsmore general then algebras of polynomials over fields. For example, V. A. Artamonov solved this problem for rings of quantum polynomials. The current state of this line of research is described in detail in the survey paper [1] containing an extensive list of references. Inmany cases, it is possible to prove that all projectivemodules of sufficiently large ranks over a given ring are free (or extended), but for projectivemodules of small ranks (for example, of rank one) this result does not hold. In this paper, we discuss problems of more special character: The problem of finding in an explicit form isomorphisms between a given projective module and a free (or extended) module, and, for an explicitly given projective module, the problem of efficient determination whether it is a free (or extended) module. By definition, a finitely generated projective module is a direct summand of a free module. Therefore, the most general way to define such a module is to indicate the idempotent matrix corresponding to the projection onto the direct summand. We show that, for modules over graded rings, the fact that a module is extended can be established if the sequence of powers of nonzero homogeneous components of the given idempotent matrix is known (Theorem 1), or under condition that certain blocks of positive homogeneous components of the same matrix vanish (Theorem 2). In the above described cases, there is a method of explicit constructing of the desired isomorphism. Statements of such kind are derived as consequences of the results of [2], where the author considered a situation which is so general that one can take as an “algebra” for which the results of [2] are valid not only linear multioperator algebras but modules over a certain class of rings. In this paper, we use the basic constructions, definitions, and notation from [2]. We make only insignificant changes in notation in order that it correspond to the generally adopted notation in the theory of rings and modules. So, the notation f1 used in [2] (for a polynomial, it corresponds to the homogeneous component of the first degree) is changed here by f0 since it corresponds to the zero component of the grading of the ring. Correspondingly, the other indices are also shifted by one.

Baer and quasi-Baer rings are important classes of algebraic objects, and their properties have roots in analysis. In this paper, we investigate rings R such that is Baer or quasi-Baer, where is either the Jacobson radical or the prime radical of R. Preliminary characterizations and results are obtained; in particular, we show that the property of being quasi-Baer is a Morita invariant.

On direct sums of Baer modules

We survey the set–theoretic methods of module theory that make it possible to equip roots of the contravariant Ext functor with filtrations built from the small roots. The power of these methods is illustrated by several applications: a solution to the Kaplansky problem on Baer modules and some of the related problems for relative Baer modules, the structure of tilting modules and classes, the structure of Matlis localizations of commutative rings, and in particular cases, proofs of the finitistic dimension conjectures, and of the telescope conjecture for module categories.

The rings of this paper are assumed to have relations of orthogonality defined on them. Such relations are uniquely determined by complete boolean algebras of ideals. Using the Stone space of these boolean algebras, and following J. Dauns and K.H. Hofmann, a sheaf-theoretic representation is obtained for rings with orthogonality relations, and the rings of global sections of these sheaves are characterized. Baer rings, f–rings and commutative semi-prime rings have natural orthogonality relations and among these the Baer rings are isomorphic to their associated rings of global sections. A special type of ideal is singled out in commutative semi-prime rings and following G. Spirason and E. Strzelecki, in an unpublished note, a characterization of a class of such rings is obtained.

Uniquely separable extensions

This paper investigates two related problems. The first is to describe the double centralizer of an arbitrary projective right iϋ-modiile. This proves to be the ring of left quotients of R with respect to a certain canonical hereditary torsion class of left ϋί-modules determined by the projective module. The second is to determine the relationship between rings of left quotients of R and S, where S is the endomorphism ring of a finitely generated projective right ϋί-module PR. It is shown that there exists an inclusion-preserving, one-to-one correspondence between hereditary torsion classes (or localizing subcategories) of left ^-modules and hereditary torsion classes of left E-modules which contain the canonical torsion class determined by PR. If QR and Qs are rings of left quotients with respect to corresponding classes, then P($$RQR is a finitely generated projective right Q^-module with Qs as its Q#-endomorphism ring. Necessary and sufficient conditions are obtained for the maximal rings of left quotients to be related in this manner. In particular, this occurs when PR is a faithful .R-module and R is either a semi-prime ring or a ring with zero left singular ideal. The situation considered includes the case where S is an arbitrary ring, SP is a left ^-generator, and R is the Sendomorphism ring of SP When SP is a projective left Sgenerator, the maximal rings of left quotients of R and S are related in the manner considered above.

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On any smooth affine surface over an algebraically closed field, it is easy to construct rank two projective modules with given determinant as the first Chern class and an arbitrary rational equivalence class of a zero cycle as the second Chern class. We will briefly indicate this method in Section 1. Here we construct projective modules over smooth affine 3-folds with prescribed Chern classes and of the right rank. That is to say, we construct rank two projective modules with prescribed first and second Chem classes and rank three projective modules with prescribed first, second and third Chern classes (Theorem 2.1). As a corollary, using Roitman's theorem on torsion in zero cycles and Suslin's cancellation theorem, one gets that a rank three projective module over a smooth affine threefold with top Chem class zero has a free direct summand of rank one (Cor. 2.4). For example, over a rational threefold any projective module splits into a free module and a rank two projective module. Thus we obtain necessary and sufficient conditions in terms of appropriate Chern classes for modules to be efficiently generated. For example, on an affine rational 3-fold any line bundle is 3-generated and any rank two projective module is 4-generated. We also prove that any maximal ideal over a smooth affine variety of dimension d is the zero of a section of a rank d projective module (Theorem 3.1). For an affine 3-fold, if A3(X) = 0, we prove the validity of the Eisenbud-Evans estimate for finitely generated modules. We do not know the answers for most of these questions in higher dimensions. Sometimes we assume that the characteristic of the ground field is different from 2, 3 and 5 due to technical reasons.**

The class of commutative rings known as Baer rings was first discussed by J. Kist [4], where many interesting properties of these rings were established. Not necessarily commutative Baer rings had previously been studied by I. Kaplansky [3], and by R. Baer himself [1]. In this note we show that commutative Baer rings, which generalize Boolean rings and p-rings, satisfy the Birkhoff conditions for a variety. Next we give a set of equations characterising this variety involving + and * as binary operations, – and as unary operations, and 0 as nullary operation. Finally we describe Baer-subdirectly irreducible commutative Baer rings and state the appropriate representation theorem.

This paper studies two dual notions in module theory—namely, radicals and socles—from the standpoint of reverse mathematics. We first consider radicals of $\\mathbb{Z}$ -modules, where the radical of a $\\mathbb{Z}$ -module $M$ is defined as the intersection of $pM=\\{px:x\\in M\\}$ with $p$ taken from all primes. It shows that $\\mathrm{ACA}_{0}$ is equivalent to the existence of radicals of $\\mathbb{Z}$ -modules over $\\mathrm{RCA}_{0}$ . We then study socles of modules over commutative rings with identity. The socle of an $R$ -module $M$ is the largest semisimple submodule of $M$ . We show that the existence of socles of modules over a commutative ring with identity is equivalent to $\\mathrm{ACA}_{0}$ over $\\mathrm{RCA}_{0}$ . Vector spaces are semisimple modules over fields. In general, semisimple modules possess nice properties of vector spaces. Lastly, we study characterizations of semisimple modules over commutative rings using techniques of reverse mathematics.