Structure of pseudo-semisimple rings

Infinite primes and unique factorization in a principal right ideal domain

Modules with chain conditions up to isomorphism

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

Louis Solomon introduced the notion of a zeta function ${\zeta _\Theta }(s)$ of an order $\Theta$ in a finite-dimensional central simple K-algebra A, with K a number field or its completion ${K_P}$ (P a non-Archimedean prime in K). In several papers, C. J. Bushnell and I. Reiner have developed the theory of zeta functions and they gave explicit formulae in some special cases. One important property of these zeta functions is the Euler product, which implies that in order to calculate ${\zeta _\Theta }(s)$, it is sufficient to consider the zeta function of local orders ${\Theta _P}$. However, since these local orders ${\Theta _P}$ are in general not principal ideal domains, their zeta function is a finite sum of so-called ’partial zeta functions’. The most complicated term is the ’genus zeta function’, ${Z_{{\Theta _P}}}(s)$, which is related to the free ${\Theta _P}$-ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e., $[A:K] = 4$). The authors mention the general case but they remark that the calculations are cumbersome. In this paper we derive an explicit method to calculate the genus zeta function ${Z_{{\Theta _P}}}(s)$ of any local hereditary order ${\Theta _P}$ in a central simple algebra over a local field. We obtain ${Z_{{\Theta _P}}}(s)$ as a finite sum of explicit terms which can be calculated with a computer. We make some remarks on the programming of the formula and give a short list of examples. The genus zeta function of the minimal hereditary orders (corresponding to the partition (1, 1, ... , 1) of n) seems to have a surprising property. In all examples, the nominator of this zeta function is a generating function for the q-Eulerian polynomials. We conclude with some remarks on a conjectured identity.

In the algebra there are several kinds of structure theorems which can be formulated without operations, using only homological tools. For instance, the wellknown fact that any universal algebra can be subdirectly embedded in a direct product of subdirectly irreducible algebras, can be formulated in a pure categorytheoretical manner. Now the question arises what its dual statement asserts. Our purpose is to give such a category which satisfies certain selfdual conditions, and making use of these, to prove structure theorems and their dual statements. The structure theorems themselves are, of course, well-known statements for algebraic stuctures. However, their duals yield some theorems of unusual type. About the possibility of the dualization there occurs some trouble. The most of the difficulties is at finding selfdual conditions being necessary to prove the theorems. So we must not make use of the condition 'every epimorphism is a normal one' which is fulfilled for groups, since its dual is false. Further the lattice of all congruencerelations of any universal algebra is a so-called compactly generated lattice. This fact plays a very important role in the proof of the theorem according to subdirect embeddings of universal algebras, nevertheless compactly generating is not a se!fdual notion. Applying the theorems proved for certain categories, we establish some particular theorems for rings, groups, modules, respectively. In w 2 we give a detailed enumeration of the usual notions and assertions of the theory of categories with respect to the importance of the dual notions and assertions, moreover, we form a system of selfdual conditions which will be satisfied by the category we are dealing with. w 3 is devoted to the investigation of subdirect embeddings, subdirect irreducibility and to the dualization of those. In w 4 we applicate the results developed before for rings, groups, module ~ and abelian groups. Most of the applications are concerned with rings.

The focus of this book is the study of the noncommutative aspects of rings and modules, and the style will make it accessible to anyone with a background in basic abstract algebra. Features of interest include an early introduction of projective and injective modules; a module theoretic approach to the Jacobson radical and the Artin-Wedderburn theorem; the use of Baer's criterion for injectivity to prove the structure theorem for finitely generated modules over a principal ideal domain; and applications of the general theory to the representation theory of finite groups. Optional material includes a section on modules over the Weyl algebras and a section on Goldie's theorem. When compared to other more encyclopedic texts, the sharp focus of this book accommodates students meeting this material for the first time. It can be used as a first-year graduate text or as a reference for advanced undergraduates.