Decomposing pairs of modules

This study wants to provide a decomposition of the module that was built into a number of special rings, such as the principal ideal domain or the Dedekind domain. The main result of this study is that the decomposition of the finitely generated module is the direct sum of the torsion module and the independent module when the ring is the principal ideal domain. When the ring is the Dedekind domain, the decomposition is the direct sum of the projective module and the torsion module. Abstrak Penelitian ini ingin memberikan dekomposisi dari modul yang dibangun Secara hingga atas beberapa ring khusus, seperti daerah ideal utama atau daerah Dedekind. Hasil utama dari penelitian ini mendapatkan dekomposisi modul yang dibangun secara hingga adalah jumlah langsung dari modul torsi dan modul bebas saat ring adalah daerah ideal utama. Saat ring adalah daerah Dedekind, didapatkan dekomposisi merupakan tambah langsung dari modul projektif dan modul torsi.

The study of pairs of modules (over a Dedekind domain) arises from two different perspectives, as a starting step in the analysis of tuples of submodules of a given module, or also as a particular case in the analysis of Abelian structures made by two modules and a morphism between them. We discuss how these two perspectives converge to pairs of modules, and we follow the latter one to obtain an alternative approach to the classification of pairs of torsionfree objects. Then we restrict our attention to pairs of free modules. Our main results are that the theory of pairs of free Abelian groups is co-recursively enumerable, and that a few remarkable extensions of this theory are decidable.

The direct sum behaviour of its projective modules is a fundamental property of any ring. Hereditary Noetherian prime rings are perhaps the only noncommutative Noetherian rings for which this direct sum behaviour (for both finitely and infinitely generated projective modules) is well-understood, yet highly nontrivial . This book surveys material previously available only in the research literature. It provides a re-worked and simplified account, with improved clarity, fresh insights and many original results about finite length modules, injective modules and projective modules. It culminates in the authors' surprisingly complete structure theorem for projective modules which involves two independent additive invariants: genus and Steinitz class. Several applications demonstrate its utility. The theory, extending the well-known module theory of commutative Dedekind domains and of hereditary orders, develops via a detailed study of simple modules. This relies upon the substantial account of idealiser subrings which forms the first part of the book and provides a useful general construction tool for interesting examples. The book assumes some knowledge of noncommutative Noetherian rings, including Goldie's theorem. Beyond that, it is largely self-contained, thanks to the appendix which provides succinct accounts of Artinian serial rings and, for arbitrary rings, results about lifting direct sum decompositions from finite length images of projective modules. The appendix also describes some open problems.

This is an introductory text on the theory of modules over rings or, equivalently, over algebras, addressed to beginning graduate students. It stresses the importance of the categorical perspective and the use of homological tools. Modules are studied both from a classical point of view and a categorical one. For this purpose, rudiments of category theory, homological algebra and representations of quivers are introduced and applied in the study of module categories. After introducing such fundamental tools as the Hom functor and the tensor product comes the study of projective, injective and flat modules. The classical results, such as the classification of modules over principal ideal domains, the Jordan-Hölder, Wedderburn-Artin, and Morita theorems, are proven, as well as the Eilenberg-Watts theorems. Radicals (of modules, algebras and categories) are thoroughly studied. Homological invariants, such as the extension and torsion modules or the homological dimensions are introduced using derived functors and are applied in module theory. The book stresses the construction and analysis of examples.

This note is devoted to the study of divisible modules over arbitrary commutative domains R with 1. Whereas divisible modules over Dedekind domains can be completely characterized by numerical invariants, not much is known about their structures in the general case. Divisible modules have been studied by Matlis [6] who was the first to distinguish between divisibility in general and h-divisibility. He established an important duality between h-divisible torsion modules and complete torsion-free R-modules [7] . Matlis [6] and Hamsher [4] characterized those domains R for which all divisible R-modules are h-divisible, e.g. by the property that the field Q of quotients of R has projective dimension 1.

Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair ⟨ R , S ⟩ \langle R,S\rangle is said to be stable if the ring of fractions of R, S − 1 R {S^{ - 1}}R , defined by S is a ring for which all finitely generated projective modules are free. For a stable pair ⟨ R , S ⟩ \langle R,S\rangle assume 2 is a unit in R and V is a free R-module of dimension ⩾ 3 \geqslant 3 . This paper examines the action of a group automorphism of G L ( V ) GL(V) (the general linear group) on the elementary matrices relative to a basis B of V. In the case that R is a local ring, a Euclidean domain, a connected semilocal ring or a Dedekind domain whose quotient field is a finite extension of the rationals, we obtain a description of the action of the automorphism on all elements of G L ( V ) GL(V) .

Let PR be a projective module with trace ideal T. An R-module XR is T-accessible if XT = X. If PR is finitely generated projective and C is the R-endomorphism ring of PR, such that CPR, then for XR, Hom (PR, XR)C is artinian (noetherian) if and only if XR satisfies the minimum (maximum) condition on T-accessible submodules. Further, if XR is T-accessible then Hom (PR, XR)C is finitely generated if and only if XR is finitely generated. The purpose of the present paper is to investigate cHom (CPR, XR), where PR is a finitely generated projective R-module and C = End (PR), the R-endomorphism ring of P, with respect to the properties of chain conditions and finite generation. Throughout this paper R is a ring with identity and all modules over R are unitary. The convention of writing module-homomorphisms on the side opposite the scalars is adopted here. 1. Preliminaries. Let PR be a finitely generated projective R-module with C = End (PR) such that CPR. The dual module of PR is (with respect to RR), RPC = Hom (CPR, RRR). It is well known, see [1], that the map PR 6P PR* = Hom (RP*, RR) given by p p, where ffp = fp, forf E P* is an R-isomorphism. LEMMA 1.1. For PR finitely generated projective, C = End (PR), the map End (PR) -End (RP*) given by c -c j, where fc= fc is a ring isomorphism. PROOF. The above map is nothing more than the composite of the following maps Hom (1, 6p) Hom (PR, PR) RHom (PR pR*)

Projective modules: Modules and homomorphisms Projective modules Completely reducible modules Wedderbum rings Artinian rings Hereditary rings Dedekind domains Projective dimension Tensor products Local rings Polynomial rings: Skew polynomial rings Grothendieck groups Graded rings and modules Induced modules Syzygy theorem Patching theorem Serre conjecture Big projectives Generic flatness Nullstellensatz Injective modules: Injective modules Injective dimension Essential extensions Maximal ring of quotients Classical ring of quotients Goldie rings Uniform dimension Uniform injective modules Reduced rank Index.

IfHis a monoid anda = u1···uk ∈ Hwith atoms (irreducible elements)u1, … ,uk, thenkis a length ofa, the set of lengths ofais denoted byⅬ(a), and ℒ(H) = {Ⅼ(a) |a ∈ H} is the system of sets of lengths ofH. LetRbe a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisorsR•can be written as a product of atoms. We show that ifRis bounded and every stably free rightR-ideal is free, then there exists a transfer homomorphism fromR•to the monoidBof zero-sum sequences over a subsetGmax(R) of the ideal class groupG(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoidsR•andBcoincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free rightR-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.

A. Fröhlich used the module index to classify the projective modules of an order in a finite dimensional commutative separable algebra over the quotient field of a Dedekind domain. This paper extends Fröhlich’s results and classifies the invertible ideals of an order in a noncommutatives eparable algebra. Several properties of invertible ideals are considered, and examples are given.