Principal Ideal Domain Research Articles

On Farrell–Tate cohomology of GL3 over rings of quadratic integers

The goal of the present paper is to push forward the frontiers of computations on mod ℓ Farrell–Tate cohomology for arithmetic groups. We deal with ℓ-rank 1 cases different from PSL2. The conjugacy classification of cyclic subgroups of order ℓ is reduced to the classification of modules of Cℓ-group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell–Tate cohomology computations, we discuss in detail the homological torsion in PGL3 over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case.

Generalized Munn rings

AbstractGeneralized Munn rings exist extensively in the theory of rings. The aim of this note is to answer when a generalized Munn ring is primitive (semiprimitive, semiprime and prime, respectively). Sufficient and necessary conditions are obtained for a generalized Munn ring with a regular sandwich matrix to be primitive (semiprimitive, semiprime and prime, respectively). Also, we obtain sufficient and necessary conditions for a Munn ring over principal ideal domains to be prime (semiprime, respectively). Our results can be regarded as the generalizations of the famous result in the theory of rings that for a ringRR,RRis primitive (semiprimitive and semiprime, respectively) if and only if so isMn(R){M}_{n}\left(R). As applications of our results, we consider the primeness and the primitivity of generalized matrix rings and generalized path algebras. In particular, it is proved that a path algebra is a semiprime if and only if it is semiprimitive.

On modules in which every finitely generated submodule is a kernel of an endomorphism

We study an R-module M in which every finitely generated submodule of M is a kernel of an endomorphism of M. Such modules are called Co-epi-finite-retractable (CEFR). We also consider CEFR condition on the co-local modules, submodules and factors of a CEFR module and direct sum of CEFR modules. Among other results, we prove that the injective hull of a simple module over a commutative Noetherian ring is uniserial if and only if it is CEFR.We investigate modules over a principal ideal ring, and show that all finitely generated torsion modules over a principal ideal domain are CEFR. Also, we show that every module over a commutative Köthe ring is CEFR. We also observe that a ring R is left pseudo morphic if and only if it is CEFR as a left R-module and we obtain some new properties of left pseudo morphic rings.

A locally Nagata PID that is not Nagata

It has previously been shown by an example of dimension one that a locally excellent integral domain need not be excellent or even Nagata. A related example showed that a locally excellent normal domain of dimension at least two need not be excellent, though that example is Nagata. Here we present a locally excellent principal ideal domain which is neither excellent nor Nagata. A second example is a two dimensional locally excellent regular ring which is Japanese but not Nagata.

Saturation of finitely-generated submodules of free modules over Prüfer domains

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).

Euclidean ideal classes in Galois number fields of odd prime degree

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least $1$ is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis holds. The aim of this paper is to show the following. Suppose that $\mathbf{K}_1$ and $\mathbf{K}_2$ are two Galois number fields of odd prime degree with cyclic class groups and Hilbert class fields that are abelian over $\mathbb{Q}$. If $\mathbf{K}_1\mathbf{K}_2$ is ramified over $\mathbf{K}_i$, then at least one $\mathbf{K}_i$ ($i \in \{1,2\}$) must have a Euclidean ideal class.

When does a quotient ring of a PID have the cancellation property?

An ideal $I$ of a commutative ring is called a cancellation ideal if $IB = IC$ implies $B = C$ for all ideals $B$ and $C$. Let $D$ be a principal ideal domain (PID), $a, b \in D$ be nonzero elements with $a \nmid b$, $(a, b)D = dD$ for some $d \in D$, $D_a = D/aD$ be the quotient ring of $D$ modulo $aD$, and $bD_a = (a,b)D/aD$; so $bD_a$ is a nonzero commutative ring. In this paper, we show that the following three properties are equivalent: (i) $\frac{a}{d}$ is a prime element and $a \nmid d^{2}$, (ii) every nonzero ideal of $bD_a$ is a cancellation ideal, and (iii) $bD_a$ is a field.

ON CYCLIC DECOMPOSITION OF Z-MODULE M_{m×r}(Z_n)

A torsion module over a principal ideal domain has special properties related to the way how it is decomposed either into primary or cyclic submodules. This paper carries out a special case of such module over the ring of integer, which consists of all matrices with entries from the set of integer modulo n. The result shows that its decomposition depends on the prime factors of n.

The Decomposition of a Finitely Generated Module over Some Special Ring

This research aims to give the decompositions of a finitely generated module over some special ring, such as the principal ideal domain and Dedekind domain. One of the main problems with module theory is to analyze the objects of the module. This research was using a literature study on finitely generated modules topics from scientific journals, especially those related to the module theory. And by selective cases we find a pattern to build a conjecture or a hypothesis, by deductive proof, we prove the conjecture and state it as a theorem. The main result in this study is the decomposition of the finitely generated module is a direct sum of the torsion submodule and torsion-free submodule. Since the torsion-free module is always a free module over a principal ideal domain, then the torsion-free submodule is a free module. Last, we generalize the ring, from a principal ideal domain, to a Dedekind domain. We found then the torsion-free submodule became a projective module. Then the decomposition of the finitely generated module is a direct sum of the torsion submodule and the projective submodule. These results should help the researchers to analyze the objects on module theory.

Linear systems of Diophantine equations

Given free modules $M\subseteq L$ of finite rank $f\geq 1$ over a principal ideal domain $R$, we give a procedure to construct a basis of $L$ from a basis of $M$ assuming the invariant factors or elementary divisors of $L/M$ are known. Given a matrix $A\in M_{m,n}(R)$ of rank $r$, its nullspace $L$ in $R^n$ is a free $R$-module of rank $f=n-r$. We construct a free submodule $M$ of $L$ of rank $f$ naturally associated with $A$ and whose basis is easily computable, we determine the invariant factors of the quotient module $L/M$ and then indicate how to apply the previous procedure to build a basis of $L$ from one of $M$.

Non-existence of graded unital homomorphisms between Leavitt algebras and their Cuntz splices

Let [Formula: see text], let [Formula: see text] be the graph consisting of one vertex and [Formula: see text] loops and let [Formula: see text] be its Cuntz splice. Let [Formula: see text] and [Formula: see text] be the Leavitt path algebras over a unital ring [Formula: see text]. Let [Formula: see text] be the cyclic group on [Formula: see text] elements. Equip [Formula: see text] and [Formula: see text] with their natural [Formula: see text]-gradings. We show that under mild conditions on [Formula: see text], which are satisfied, for example, when [Formula: see text] is a field or a principal ideal domain, there are no unital [Formula: see text]-graded ring homomorphisms [Formula: see text] nor in the opposite direction.

Linear Diophantine equations in several variables

Let R be a ring and let (a1,…,an)∈Rn be a unimodular vector, where n≥2 and each ai is in the center of R. Consider the linear equation a1X1+⋯+anXn=0, with solution set S. Then S=S1+⋯+Sn, where each Si is naturally derived from (a1,…,an), and we give a presentation of S in terms of generators taken from the Si and appropriate relations. Moreover, under suitable assumptions, we elucidate the structure of each quotient module S/Si. Furthermore, assuming that R is a principal ideal domain, we provide a simple way to construct a basis of S and, as an application, we determine the structure of the quotient module S/Ui, where each Ui is a specific module containing Si.

Arithmetic invariants of pseudoreflection groups and regular graded algebras

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudoreflection groups over Dedekind domains. In the special case of a principal ideal domain in which the group order is invertible it is proved that this ring of invariants is isomorphic to a polynomial ring. An intermediate result is that every finitely generated regular graded algebra over a Dedekind domain is isomorphic to a tensor product of blowup algebras.

The cohomology rings of homogeneous spaces

Let G be a compact connected Lie group and K a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of G and K is invertible in a given principal ideal domain k. It is known that in this case the cohomology of the homogeneous space G / K with coefficients in k and the torsion product of H ∗ ( B K ) and k over H ∗ ( B G ) are isomorphic as k-modules. We show that this isomorphism is multiplicative and natural in the pair ( G , K ) provided that 2 is invertible in k. The proof uses homotopy Gerstenhaber algebras in an essential way. In particular, we show that the normalized singular cochains on the classifying space of a torus are formal as a homotopy Gerstenhaber algebra.

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

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.

N Sequence Prime Ideals

In this paper, the concepts of -sequence prime ideal and -sequence quasi prime ideal are introduced. Some properties of such ideals are investigated. The relations between -sequence prime ideal and each of primary ideal, -prime ideal, quasi prime ideal, strongly irreducible ideal, and closed ideal, are studied. Also, the ideals of a principal ideal domain are classified into quasi prime ideals and -sequence quasi prime ideals.

Canonical form of Involutory Matrices over the Domain of Principal Ideals with Respect to Similarity Transformations

Canonical form of Involutory Matrices over the Domain of Principal Ideals with Respect to Similarity Transformations

The ultradiscrete Toda lattice and the Smith normal form of bidiagonal matrices

The discrete Toda lattice preserves the eigenvalues of tridiagonal matrices, and convergence of dependent variables to the eigenvalues can be proved under appropriate conditions. We show that the ultradiscrete Toda lattice preserves invariant factors of a certain bidiagonal matrix over a principal ideal domain and prove convergence of dependent variables to invariant factors using properties of the box and ball system. Using this fact, we present a new method for computing the Smith normal form of a given matrix.

More on Prime, Maximal and Principal Soft Ideals of Soft Rings

In this paper, we aim to extend the studies [M. R. Alimoradi, R. Rezaei and M. Rahimi, Some notes on ideals in soft rings, Journal Australian Journal of Basic and Applied Sciences 6(3) (2012) 717–721; F. Koyuncu and B. Tanay, Some soft algebraic structures, Journal of New Results Science 10(2016) 38–51] as regards maximal, prime and principal soft ideal of soft rings, characterize soft rings with these soft ideals and also provide some more relations between maximal, prime and principal soft ideals of soft rings. The notions of maximality and primeness points of soft ideals of a soft rings are defined, maximal and prime idealistic soft rings as well as maximal, prime and principal soft ideal of soft rings and their basic properties are more investigated under certain conditions, especially by means of homomorphism and epimorphism of rings. We apply some of the basic results about maximal ideals and prime ideals in classical abstract algebra to maximal, prime and principal idealistic soft rings and we give some of their interrelations between each others.

On homology of Lie algebras over commutative rings

We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over Z, and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie algebra which is flat as a module. As an auxiliary result we prove that the Koszul complex of a module M over a principal ideal domain that connects the exterior and the symmetric powers 0→ΛnM→M⊗Λn−1M→…→Sn−1M⊗M→SnM→0 is purely acyclic.