Generic Coordinate Systems in Two Variables Over a Principal Ideal Domain

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.

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.

An almost prime submodule is a generalization of prime submodule introduced in 2011 by Khashan. This algebraic structure was brought from an algebraic structure in ring theory, prime ideal, and almost prime ideal. This paper aims to construct similar properties of prime ideal and almost prime ideal from ring theory to module theory. The problem that we want to eliminate is the multiplication operation, which is missing in module theory. We use the definition of module annihilator to bridge the gap. This article gives some properties of the prime submodule and almost prime submodule of CMS module over a principal ideal domain. A CSM module is a module that every cyclic submodule. One of the results is that the idempotent submodule is an almost prime submodule.

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

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].

We propose to give an algorithm for computing the $R$-saturation of a finitely-generated submodule of a free module $E$ over a Prüfer domain $R$. To do this, we start with the local case, that is, the case where $R$ is a valuation domain. After that, we consider the global case ($R$ is a Prüfer domain) using the dynamical method. The proposed algorithm is based on an algorithm given by Ducos, Monceur and Yengui in the case $E=R[X]^m$ which is reformulated here in a more general setting in order to reach a wider audience. The last section is devoted to the case where $R$ is a Bézout domain. Particular attention is paid to the case where $R$ is a principal ideal domain ($\mathbb{Z}$ as the main example).

The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell complexes and introduce the notion of regular chain complexes. In the case of chain complexes coming from simplicial complexes we recover the classical notions but, in contrast to the topological case, in the abstract setting shellings turn out to be a weaker homological invariant. In particular, we study special chain complexes, which are cones, and a class of regular chain complexes, which we call totally regular, for which we can obtain complete homological information if an additional condition is fulfilled.

25 Homological Algebra over a Principal Ideal Domain: Kuuml;nneth and Universal Coefficient Theorems

Let Mn and Nn be closed manifolds, and let G be any (nonzero) module. (1) If f: M3 -N3 is C3 G-acyclic, then there is a closed C3 3-manifold K3 such that N3 # K3 is diffeomorphic to M3, and f1(y) is cellular for all but at most r points y E N3, where r is the number of nontrivial G-cohomology 3-spheres in the prime decomposition of K3. (2) Iff: M3 _> M3 orf: S3 _* M3 is G-acyclic, then f is cellular. In case G is Z or Z, (p prime), results analogous to (1) and (2) in the topological category have been proved by Alden Wright. (3) If f: Mn -* Mn or f: Sn * Mn is real analytic monotone onto, then f is a homeomorphism. 1. Conventions and definitions. Let G be a module, and let A c Mn be compact. Then A is acyclic (k-acyclic; cellular) is the reduced Cech cohomology module H*(A; G)=0 (Hk(A;G)=O; there are n-cells AkcMMn such that Ak+lcint Ak and nk Ak=A). A proper onto map f: Mn -Nn is monotone (acyclic; cellular) if, for each y E Nn, f -1(y) is connected (acyclic; cellular). The branch set Bf is the set of points in Mn at which f fails to be a local homeomorphism. Standing hypotheses. Unless otherwise specified, all manifolds are connected, separable, and without boundary. Whenever the statement of a theorem refers to a map f without specifying its domain and range, it is understood that f: Mn -Nn is proper and onto. Whenever coefficients of cohomology are not specified, any (nonzero) module G may be used. Other conventions and definitions are as in [6, pp. 185-186]. For other work on monotone, acyclic, and cellular mapsf: Mn Nn, see bibliographies of this paper, [1], [15], and (vast) [30]. 2. Real analytic monotone maps. 2.1. DEFINITION. Let G be a module, and let Cf = Cf(G) = f I(Cl {y E Nn: fl*(f 1(y); G) # 0}). 2.2. LEMMA. Let L be a principal ideal domain, let Mn be orientable over L, and let dim (f(Cf)) < 0. Received by the editors August 26, 1970. AMS 1970 subject classifications. Primary 57A60, 54C10; Secondary 57D35, 32C05.