On the Schröder–Bernstein property for modules

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).

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.

Poor modules with no proper poor direct summands

A right R-module M is virtually extending (or CIS) if every complement submodule of M is isomorphic to a direct summand of M, and M is called a virtually C2-module if every complement submodule of M which is isomorphic to a direct summand of M is itself a direct summand. The class of virtually extending modules (respectively, virtually C2-modules) is a strict and simultaneous generalization of extending modules (respectively, unifies extending modules and C2-modules): M is a semisimple module if and only if M is virtually semisimple and C2, and M is an extending module if and only if M is virtually extending and virtually C2. Furthermore, every virtually simple right R-module is injective if and only if R is a right V-ring and the class of virtually simple right R-modules coincides with the class of simple right R-modules. Among other results, we show that (1) if all cyclic sub-factors of a cyclic weakly co-Hopfian right R-module M are virtually extending, then M is a finite direct sum of uniform submodules; (2) every distributive virtually extending module over any Noetherian ring is a direct sum of uniform submodules; (3) over a right Noetherian ring, every virtually extending module satisfies the Schröder-Bernstein property; (4) being virtually extending (VC2) is a Morita invariant property; (4) if is a VC2-module where denotes the injective hull, then M is injective. Communicated by Scott Chapman

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.

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 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.

In 2002, Anderson and Dumitrescu introduced [Formula: see text]-Noetherian rings, a significant advancement in ring theory. Later, in 2018, Bennis and El Hajoui expanded on this by introducing two new classes of rings that generalize the concept of coherent rings: [Formula: see text]-coherent rings, where every finitely generated ideal is [Formula: see text]-finitely presented, and [Formula: see text]-[Formula: see text]-coherent rings, where every [Formula: see text]-finite ideal is [Formula: see text]-finitely presented. This paper builds on these concepts to introduce the [Formula: see text]-[Formula: see text]-(weak) global dimension for rings with a multiplicative subset [Formula: see text], offering a new perspective on ring structures. We also explore [Formula: see text]-[Formula: see text]-variants of specific ring types, including semisimple, von Neumann, and (semi)hereditary rings. Finally, we extend the generalization of [Formula: see text]-rings and [Formula: see text]-coherent rings, originally introduced by Costa in 1994, to rings with a multiplicative subset [Formula: see text], deepening the understanding of ring theory.

The class of rings studied in this paper properly contains the class of right distributive rings which have at least one completely prime ideal in the Jacobson radical. Amongst other results we study prime and semiprime ideals, right noetherian rings with comparability and prove a structure theorem for rings with comparability. Several examples are also given.

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.

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.

Noetherian rings with many derivations

Residual coordinates over one-dimensional rings

Let A be a noetherian commutative ring which is a homomorphic image of a Gorenstein ring.The first result in this note is that A is a Cohen-Macaulay ring if and only if every finitely generated /I-module can be embedded in a finitely generated Amodule T which is pointwise of finite injective dimension (i.e.all localizations of 7"are of finite injective dimension).The second result is that if A is Cohen-Macaulay, then every finitely generated Amodule can be embedded in a finitely generated /(-module of finite projective dimension if and only if A is Gorenstein.Let A denote a commutative noetherian ring.As a running hypothesis we will assume that A is a homomorphic image of a Gorenstein ring (a commutative noetherian ring R is Gorenstein if Rv has finite injective dimension as an /?p-module, for every prime ideal p of R; see [2] for other definitions equivalent to this one).However, this class of noetherian rings is rather extensive, since it contains most of the rings occurring in algebraic geometry, and, in particular, it contains the complete local rings, by Cohen's structure theorem [5].All modules will be finitely generated (f.g.), and for a f.g.A-module M and p g Spec A, depth Mp denotes (as usual) the maximal length of a Mpregular sequence in pAp, while grade M is the maximal length of an /(-regular sequence in Ann M. Furthermore, recall that a f.g.nonzero Amodule Q is said to be a Gorenstein module if Í2 pointwise is of finite injective dimension (i.e.inj dim Op< co for all p e Supp O) and depth Í2p= depth Ap for all p e Supp O. Since A is a homomorphic image of a Gorenstein ring, by assumption, there exists a Gorenstein /1-module which is of rank one (i.e.Hom(i2, D.) = A), if A is a Cohen-Macaulay ring (cf.Sharp [II], and see [10], [6] and [7] for further properties of Gorenstein modules).Theorem I.A is a Cohen-Macaulay ring if and only if every f.g.Amodule can be embedded in a f.g.A-module which pointwise is of finite injective dimension.

A ring R has the (A)-property (resp., strong (A)-property) if every finitely generated ideal of R consisting entirely of zero divisors (resp., every finitely generated ideal of R generated by a finite number of zero-divisors elements of R) has a nonzero annihilator. The class of commutative rings with property (A) is quite large; for example, Noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose total ring of quotients are von Neumann regular. Let f : A → B be a ring homomorphism and J be an ideal of B. In this paper, we investigate when the (A)-property and strong (A)-property are satisfied by the amalgamation of rings denoted by A ⋈fJ, introduced by D'Anna, Finocchiaro and Fontana in [3]. Our aim is to construct new original classes of (A)-rings that are not strong (A)-rings, (A)-rings that are not Noetherian and (A)-rings whose total ring of quotients are not Von Neumann regular rings.