A note on almost prime submodule of CSM module over principal ideal domain

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.

The importance of principal ideal domains (PIDs), both in algebra itself and elsewhere in mathematics is undisputed. By contrast, Bezout rings, ‡ although they represent a natural generalization of PIDs, play a much smaller role and are far less well known. It is true that many of the properties of PIDs are shared by Bezout rings, but the practical value of this observation is questioned by many on the grounds that most of the Bezout rings occuring naturally are in fact PIDs. However, there are several fairly natural methods of constructing Bezout rings from other rings, leading to wide classes of Bezout rings which are not PIDs, and it is the object of this paper to discuss some of these methods.

Recall that a commutative ring R is said to be a divided ring if its each prime ideal P is comparable with each principal ideal (a), where $$a\in R.$$ In this paper, we extend the notion of divided rings to modules in two different ways: let R be a commutative ring with identity and M a unital R-module. Then M is said to be a divided (weakly divided) module if its each prime submodule N of M is comparable with each cyclic submodule Rm (rM) of M, where $$m\in M$$ ($$r\in R)$$. In addition to give many characterizations of divided modules, some topological properties of (quasi-) Zariski topology of divided modules are investigated. Also, we study the divided property of trivial extension $$R\ltimes M$$.

Given a square matrix B over a principal ideal domain D and an ideal J of D, the J-ideal of B consists of the polynomials f∈D[X] such that all entries of f(B) are in J. It has been shown that in order to determine all J-ideals of B it suffices to compute a generating set of the (pt)-ideal of B for finitely many prime powers pt. Moreover, it is known that a (pt)-ideal is generated by a set of polynomials of the form pt−sνs for certain s≤t where each νs is a monic polynomial of minimal degree in the (ps)-ideal of B. However, except for the case of diagonal matrices, it was not known how to determine these polynomials explicitly. We present an algorithm which allows us to compute the polynomials νs for general square matrices. Exploiting one of McCoy's theorems we first compute some set of generators of the (ps)-ideal of B which then can be used to determine νs.

This correspondence is addressed to the question of invertibility of linear dynamical systems. Specifically, inversion over principal ideal domains is discussed in view of the fact that 2-D systems can be considered as 1-D systems over a principal ideal domain.

We consider a question raised by Mowaffaq Hajja about the structure of a principal ideal domain R having the property that 1 is the only unit of R. We also examine this unit condition for the case where R is a Euclidean domain. We prove that a finitely generated Euclidean domain having 1 as its only unit is isomorphic to the field with two elements F 2 or to the polynomial ring F 2[X]. On the other hand, we establish existence of finitely generated principal ideal domains R such that 1 is the only unit of R and R is not isomorphic to F 2 or to F 2[X]. We also construct principal ideal domains R of infinite transcendence degree over F 2 with the property that 1 is the only unit of R. *Part of this work was prepared while M. Roitman enjoyed the hospitality of Purdue University.

In this chapter we introduce and study the algebra U(G) of operators affiliated to N(G) for a group G. A G-operator f: dom(f) ⊂ V → W of Hilbert N(G)-modules is an operator whose domain dom(f) is a linear G-invariant subspace and which satisfies f(gx) = gf(x) for all x ∈ dom(f) and g ∈ G. The algebra U(G) consists of densely defined closed G-operators a: dom(a) ⊂ l 2(G) → l 2(G) and contains N(G) as a subalgebra. It is constructed in such a way that an element f : l 2(G) → l 2(G) in N(G) is a weak isomorphism if and only it is invertible in U(G). This is reflected algebraically by the fact that U(G) is the Ore localization of N(G) with respect to the set of non-zero divisors (see Theorem 8.22 (1)). It does not come with a natural topology anymore but has nice ring theoretic properties. Namely, U(G) is von Neumann regular, i.e. for any r ∈ U(G) there is s ∈ U(G) with rsr = r, or, equivalently, any finitely generated submodule of a projective U(G)-module is a direct summand (see Theorem 8.22 (3)). We have already mentioned in Example 6.12 that N(G) behaves in several ways like a principal ideal domain except that N(G) is not Noetherian and has zero-divisors if G is infinite. Any principal ideal domain R has a quotient field F, and in this analogy U (G) should be thought of as F. Recall that the (extended) dimension of an arbitrary R-module M is the same as the F-dimension of the F-vector space F ⊗ R M (see Example 6.12).

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.

Feedback invariants for linear dynamical systems over a principal ideal domain

Let D be a principal ideal domain (PID), I be an ideal of D, and X be an indeterminate over D. Let [D;I][X] be the subring of the power series ring consisting of all power series in such that for all large i. By definition, the polynomial ring and the power series ring are special cases of when and I = D, respectively. In this article, we investigate the ring in the case I is a nonzero proper ideal of D. We prove that R is a two-dimensional non-Noetherian ring. For each maximal ideal P of D, it is shown that is a height-one prime ideal of R. The set of units of R is given and the spectrum of R is also described. Unlike the power series ring the ring R is not a unique factorization domain (UFD). Furthermore, when I is a nonzero prime ideal, R does not satisfy both ACCP and the atomic property. In obtaining results on R, we introduce and sometimes use results on the ring RS , where I = dD with and Closely related to R, the ring RS is shown to be a Noetherian UFD with Krull dimension at most two. Moreover, RS has Krull dimension two exactly when I is not contained in the Jacobson radical of D; otherwise RS is a PID.

for all ~, t E L . In particular 2~(r162 ~ ( ~ ) ) = 0 for all ~ e L . This is the natural integral generalization of the orthogonal Lie algebra ; see JACOBSON [1, p. 6]. A similar definition where ~ is sympleetic, instead of symmetric, gives the symplectie Lie algebras. Thus we include as special cases many of the classical Lie algebras. We shall investigate the situation analogous to orthogonal groups; tha t is, we will give necessary and sufficient conditions for a mapping ~v : J -~ L (J a submodule of L) to extend to an element of s (r L, R). We assume throughout tha t 2 # 0 and shall investigate the following three cases : R a field, a valuation ring and a principal ideal domain. The situation for orthogonal Lie algebras is much simpler than for orthogonal groups; in fact little is known when R is a principal ideal domain and our results here might be viewed as a linear approximation. For any 2,/~ 9 L the mapping u #: L -+ L defined by

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

We consider the secure computation problem in a minimal model, where Alice and Bob each holds an input and wish to securely compute a function of their inputs at Carol without revealing any additional information about the inputs. For this minimal secure computation problem, we propose a novel coding scheme built from two steps. First, the function to be computed is expanded such that it can be recovered while additional information might be leaked. Second, a randomization step is applied to the expanded function such that the leaked information is protected. We implement this expand-and-randomize coding scheme with two algebraic structures—the finite field and the modulo ring of integers, where the expansion step is realized with the addition operation and the randomization step is realized with the multiplication operation over the respective algebraic structures.

In what circumstances does a given group carry a Polish group topology? (A metric, separable, complete topology is called Polish, and a topology is a group if multiplication and inverse are continuous.) Two natural interpretations of this question can be considered.

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings, maximal rings, etc. Inspired by LIP, lifting properties were also defined for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congruence distributive universal algebras, etc. In this paper, we define a lifting property (LP) in commutative coherent integral quantales, structures that are a good abstraction for lattices of ideals, filters and congruences. LP generalizes all the lifting properties existing in the literature. The main tool in the study of LP in a quantale A is the reticulation of A, a bounded distributive lattice whose prime spectrum is homeomorphic to the prime spectrum of A. The principal results of the paper include a characterization theorem for quantales with LP and a characterization theorem for hyperarchimedean quantales.