On direct sums of Baer modules

The Baer and the quasi-Baer properties of rings are extended to a module theoretic setting in this chapter. Using the endomorphism ring of a module, the notions of Baer, quasi-Baer, and Rickart modules are introduced and studied. Similar to the fact that every Baer ring is nonsingular, we shall see that every Baer module satisfies a weaker notion of nonsingularity of modules (\(\mathcal{K}\)-nonsingularity) which depends on the endomorphism ring of the module. Strong connections between a Baer module and an extending module will be observed via this weak nonsingularity and its dual notion. It is shown that an extending module which is \(\mathcal{K}\)-nonsingular is precisely a \(\mathcal{K}\)-cononsingular Baer module. This provides a module theoretic analogue of the Chatters-Khuri theorem for rings. Direct summands of Baer and quasi-Baer modules respectively inherit these properties. This provides a rich source of examples of Baer and quasi-Baer modules, since one can readily see that for any (quasi-)Baer ring R and an idempotent e in R, the right R-module eR R is always a (quasi-)Baer module. It will be seen that every projective module over a quasi-Baer ring is a quasi-Baer module. Connections of a (quasi-)Baer module and its endomorphism ring are discussed. Characterizations of classes of rings via the Baer property of certain classes of free modules over them are presented. An application also yields a type theory for \(\mathcal{K}\)-nonsingular extending (continuous) modules which, in particular, improve the type theory for nonsingular injective modules provided by Goodearl and Boyle.

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.

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

A necessary and sufficient condition is given for a semilocal domain to have the property that every finitely generated module is a direct sum of cyclic modules. Recent work by Tom Shores and Roger Wiegand [SW] and [SWB] has shed new light on the old question: What is the class of commutative rings for which every finitely generated module is a direct sum of cyclic submodules? Following their terminology, such a ring is called an FGC-ring. In this paper, we are able to settle one of their conjectures [SWB, p. 1279], thus giving a characterization of reduced FGC-rings with Noetherian maximal ideal spectrum. We show that, for R a reduced ring with Noetherian maximal ideal spectrum, R is an FGC-ring and only R is a finite direct sum of h-local Bezout domains and each localization of R is an almost maximal valuation ring. The if part of the statement above was shown in [SW] to be an easy consequence of results due to Eben Matlis [M]. Other known results on FGC-rings are reviewed in [SW]; the reader is also referred to [SW] for definitions of the concepts above. The examples constructed in [W] show that our main result answers the FGC question for a large class of rings where it was unknown. Of course it would be nice to give a complete characterization of reduced FGC-rings. This would require settling another question in [SW]: Does every FGC-ring have Noetherian maximal ideal spectrum? They show the answer is yes the ring has fewer than 2' prime ideals [SW, Corollary 6.8]. 1. The main theorem. The result stated in the introduction will easily follow from the Theorem below: Theorem. Let R be an FGC-domain. Then every nonzero prime ideal P is contained in a unique maximal ideal. Proof. Suppose R contains two distinct maximal ideals M and N with Received by the editors May 16, 1974. AMS (MOS) subject classifications (1970). Primary 13C05, 13F05; Secondary 13A15. Copyright ? 1975, Anierican Mathematical Society

We study an R-module M such that for every proper finitely generated submodule K of M, Hom R ( M / K , M ) ≠ 0 . Such modules are called co-finite-retractable (simply CFR). We consider when factor modules, direct sums and direct summands of CFR modules are CFR. Also, we investigate the injective hull of CFR modules. We explore certain properties of a ring R such that every (cyclic, free) module is CFR. Also, we show that R is a semisimple ring if and only if every R-module is regular if and only if every CFR module over R is regular.

Free left modules over right Bezout domains are characterized by a set of geometric axioms resembling those of U. Brehm, where results of this kind have been obtained for torsion-free left modules over left Ore domains. Discourses by S. E. Schmidt and the present author have come to similar results for other classes of rings and modules.

Let $V$ be a submodule of a direct sum of some elements in $\mathcal{U}$, and $X$ be a submodule of a direct sum of some elements in $\mathcal{N}$, where $\mathcal{U}$ and $\mathcal{N}$ are families of $R$-modules. A $\mathcal{U}$-free module is a generalization of a free module. According to the definition of $\mathcal{U}$-free module, we define three kinds of projective$_{\mathcal{U}}$ in this research, i.e., projective$_{\underline{\mathcal{U}}}$, projective$_{\mathcal{U}}$ module, and strictly projective$_{\mathcal{U}}$ module. The notion of strictly projective$_{\mathcal{U}}$ is a generalization of the projective module. In this research, we discuss the relationship between projective modules and the three types of modules. Furthermore, we show that the properties of $\mathcal{U}$ impact the properties of the projective$_{\mathcal{U}}$ module so that we can determine some properties of the projective$_{\mathcal{U}}$ module based on the properties of the family of $\mathcal{U}$ of $R$-modules.

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.

In this paper, we introduce and study the properties of chain conditions on M-cyclic submodules defined as M-Noetherian and M-Artinian modules and rings. We prove that the property of being iso-M-Noetherian (iso-M-Artinian) is inherited by submodules, quotient modules, and finite sums as well as finite direct sums. Finally, we prove the Hilbert basis theorem for iso-M-Noetherian (iso-M-Artinian) and verify that the property of modules of being M-Noetherian (M-Artinian) and iso-M-Noetherian (iso-M-Artinian) are Morita invariant.

In this paper we take a natural homomorphism f M : U Hom R ( M , N ) → HomU ( UM , U N ) where U is an R-Algebra , R=CenU and M , N are any two R-modules.Here we have shown that 1. If P is any projective module then f P is a monomorphism. 2. If G be any generator then f ) ( n G is an epimorphism. 3. If M be a generator and N is any free module then f ) (M TrN is always a monomorphism. 4. If M is a generator and N is any finitely generated free R-module then f ) (M TrN is an isomorphism. 5. If M be any artinian module such that it is subdirect product of simple module then f M is monomorphism iff for each submodule K , f K is monomorphism. 6. Let M be a generator and N is any free module if U is finitely presented then f ) (M TrN is an isomorphism. 7. Let M be any left R-module such that f ) ( Re / N j M M is an isomorphism then f M is an isomorphism.

In 1982 Richard P. Stanley conjectured that any finitely generated \Bbb Zn-graded module M over a finitely generated \Bbb Nn-graded \Bbb K-algebra R can be decomposed as a direct sum M e ⊕i e 1t νiSi of finitely many free modules νiSi which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least depth M. We will study this conjecture for modules M e R/I, where R is a polynomial ring and I a monomial ideal. In particular, we will prove that Stanley's Conjecture holds for the quotient modulo any generic monomial ideal, the quotient modulo any monomial ideal in at most three variables, and for any cogeneric Cohen-Macaulay ring. Finally, we will give an outlook to Stanley decompositions of arbitrary graded polynomial modules. In particular, we obtain a more general result in the 3-variate case.

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.

Harada (cf. 1983) introduced the lifting property for maximal submodules and the extending property for simple submodules, and Oshiro (cf. 1983b) definitely introduced lifting modules and extending modules. Since then, the method and results in their works urged not only the study on (quasi-)discrete and (quasi-)continuous modules but also the one on themselves. The reader is referred to Mohamed and Müller (1990) and Dung et al. (1994) for the research on lifting modules and extending modules. In this article, we study the following fundamental open problems: When is a direct sum ⨁IMi of lifting (extending) modules {Mi}I lifting (extending)? These problems are unsolved even in the case that the index set I is finite. Now, the purpose of this article is to study the problem concerned with finite direct sums of lifting modules.

Let R be an integral domain, K its field of fractions, and M n (R) the R-algebra of square matrices of order n and entries from R. In this paper we present two ways of constructing space-time block codes as submodules of M n (R) for n ≥ 1: first, by embedding free associative R-algebras of finite rank n with no zero divisors into M n (R); second, by injecting free R-modules of finite rank n into M n (R) ∩ GL n (K) ∪{ 0}. Some examples of such space-time block codes are given.

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.**