Principal Ideal Ring Research Articles

Sagbi bases over the product of rings

Let $R$ be a commutative ring such that $R=R_1\times \cdots \times R_n$. In this paper, we give a method to compute (strong) Sagbi bases for subalgebras of a polynomial ring over $R$ provided that these bases are computable in a polynomial ring over $R_i$ for $1\leq i \leq n$. As an application, we prove the existence of strong Sagbi bases for subalgebras in a polynomial ring over a principal ideal ring.

MacWilliams Identities and Generator Matrices for Linear Codes over ℤp4[u]/(u2 − p3β, pu)

Suppose that R=Zp4[u] with u2=p3β and pu=0, where p is a prime and β is a unit in R. Then, R is a local non-chain ring of order p5 with a unique maximal ideal J=(p,u) and a residue field of order p. A linear code C of length N over R is an R-submodule of RN. The purpose of this article is to examine MacWilliams identities and generator matrices for linear codes of length N over R. We first prove that when p≠2, there are precisely two distinct rings with these properties up to isomorphism. However, for p=2, only a single such ring is found. Furthermore, we fully describe the lattice of ideals of R and their orders. We then calculate the generator matrices and MacWilliams relations for the linear codes C over R, illustrated with numerical examples. It is important to address that there are challenges associated with working with linear codes over non-chain rings, as such rings are not principal ideal rings.

When do modules mimic arbitrary sets?

We study rings whose modules and module homomorphisms display behavior similar to that of sets and their maps. For example, whenever there is an epimorphism A → B , there is a monomorphism B → A (Artinian principal ideal rings (PIR) satisfy this property and its dual for all modules). As a byproduct of this framework, we prove that a ring every factor ring of which cogenerates its cyclic right modules (one-sided version of Kaplansky’s dual rings) is right Artinian and right serial. Consequently, R is an Artinian PIR if and only if every factor ring of R cogenerates its finitely generated right modules. These results can be viewed as partial answers to the CF problem, the FGF problem due to Faith and a question of Faith and Menal on strongly Johns rings. Some known results and the above one yield the following: A ring R is a direct sum of right Artinian right chain rings and Artinian PIR’s if and only if every factor ring of R cogenerates its (uniform) cyclic right modules (with nonzero socle); so, such rings coincide with the right CES-rings of Jain and Lopez-Pérmouth, rings whose factors are right CF and rings that satisfy the above mentioned property for their cyclic right modules A and B. Finally, a ring is either simple Artinian or a right Artinian right chain ring if and only if one of any two cyclic right modules embeds in the other.

Solving systems of algebraic equations over finite commutative rings and applications

AbstractSeveral problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptosystems. A finite chain ring is a finite ring admitting exactly one maximal ideal and every ideal being generated by one element. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residue field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gröbner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Plücker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.

Ideal factoriality, semistar operations, and quasiprincipal Ideals

The w-operation, a “smoother” variant of the classic t-operation on the set of ideals of a commutative ring R, has recently attracted significant attention. We study when R’s monoid of w-ideals is “factorial” in some sense. For example, we show that every proper w-ideal of R has a unique up to order irredundant w-factorization into w-unfactorable w-ideals if and only if R is a finite direct product of Krull domains and principal ideal rings, if and only if every w-ideal is w-quasiprincipal. We prove several analogous results regarding other kinds of “w-ideal factoriality” and generalize these results via semistar operations.

On Linear Codes over Finite Singleton Local Rings

The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In this paper, we present and successfully establish a new approach for linear codes of any finite length over local rings that are not necessarily chains. The main focus of this study is to produce generating characters, MacWilliams identities and generator matrices for codes over singleton local Frobenius rings of order 32. To do so, we first start by characterizing all singleton local rings of order 32 up to isomorphism. These rings happen to have strong connections to linear binary codes and Z4 codes, which play a significant role in coding theory.

Spectral equivalence of smooth group schemes over principal ideal local rings

Let G be a smooth linear group scheme of finite type. For any positive integer k and a finite field F, let Wk(F) be the ring of Witt vectors of length k over F. We show that the group algebras of G(F[t]/(tk)) and G(Wk(F)) are isomorphic (i.e. the multi-sets of the dimensions of the irreducible representations are equal) for any positive integer k and finite field F with large enough characteristic. We also prove that if charF is large enough, then the cardinality of the set {dim⁡ρ|ρ∈irr(G(F))} is bounded uniformly in F.

On the rank decoding problem over finite principal ideal rings

The rank decoding problem has been the subject of much attention in this last decade. This problem, which is at the base of the security of public-key cryptosystems based on rank metric codes, is traditionally studied over finite fields. But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings.In this paper, we show that solving the rank decoding problem over finite principal ideal rings is at least as hard as the rank decoding problem over finite fields. We also show that computing the minimum rank distance for linear codes over finite principal ideal rings is equivalent to the same problem for linear codes over finite fields. Finally, we provide combinatorial type algorithms for solving the rank decoding problem over finite chain rings together with their average complexities.

IDEALS AND ALMOST PRINCIPAL IDEALS IN EUCLIDEAN Γ−SEMIRINGS

The generalization of the ring of ordinary integers and their properties into an Euclidean ring is well known. Every ideal in an Euclidean ring is a principal ideal, as is also widely known. That is, the Euclidean ring is a principal ideal ring. This paper aims to generalize the Γ−semiring of non-negative integers and their properties by defining Euclidean Γ−semiring. A Euclidean Γ−semiring is one of the many special classes of Γ−semirings. Finally, the special class of Γ−semirings discussed in this paper is the class of almost principal ideal Γ−semirings.

Commutative rings whose proper ideals decompose into local modules

For a commutative local ring R, we know that every ideal is a direct sum of local modules if and only if the unique maximal ideal of R is of the form , where and are principal ideal rings and for all γ’s. This motivates us to determine a commutative ring, not necessarily local, in which every (proper) ideal is a direct sum of local modules. We prove that such a ring is exactly a finite direct product of local rings R with the unique maximal ideal in the above mentioned form. Moreover, we characterize more precisely certain commutative rings such as Noetherian rings, reduced rings, Rickart rings, semi-Artinian rings and self-injective rings, in which every (proper) ideal is a direct sum of local modules.

On sum annihilator ideals in Ore extensions

A ring $R$ is called a left Ikeda-Nakayama ring (left IN-ring) if the right annihilator of the intersection of any two left ideals is the sum of the two right annihilators. As a generalization of left IN-rings, a ring $R$ is called a right SA-ring if the sum of right annihilators of two ideals is a right annihilator of an ideal of $R$. It would be interesting to find conditions under which an Ore extension $R[x; \alpha, \delta]$ is IN and SA. In this paper, we will present some necessary and sufficient conditions for the Ore extension $R[x;\alpha, \delta]$ to be left IN or right SA. In addition, for an $(\alpha,\delta)$-compatible ring $R$, it is shown that: (i) If $S = R[x;\alpha,\delta]$ is a left IN-ring with ${\rm{Idm}}(R) ={\rm{Idm}}(R[x;\alpha, \delta])$, then $R$ is left McCoy. (ii) Every reduced left IN-ring with finitely many minimal prime ideals is a semiprime left Goldie ring. (iii) If $R$ is a commutative principal ideal ring, then $R$ and $R[x]$ are IN. (iv) If $R$ is a reduced ring and $n$ is a positive integer, then $R$ is right SA if and only if $R[x]/(x^{n+1})$ is right SA.

Left co-Köthe rings and their characterizations

Köthe’s classical problem posed by G. Köthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe’s problem for basic finite-dimensional algebras. But, so far, Köthe’s problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left Köthe rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe’s problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe rings, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them.

Commutative rings whose proper ideals are pure-semisimple

Recall that an R-module M is pure-semisimple if every module in the category is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that “a commutative ring R is pure-semisimple (i.e., every R-module is a direct sum of finitely generated modules) if and only if every R-module is a direct sum of cyclic modules, if and only if, R is an Artinian principal ideal ring”. Consequently, every (or finitely generated, cyclic) ideal of R is pure-semisimple if and only if R is an Artinian principal ideal ring. Therefore, a natural question of this sort is “whether the same is true if one only assumes that every proper ideal of R is pure-semisimple?” The goal of this paper is to answer this question. The structure of such rings is completely described as Artinian principal ideal rings or local rings R with the maximal ideals which Rx is Artinian uniserial and T is semisimple. Also, we give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple.

Restricted iso-minimum condition

Abstract A restricted artinian ring is a commutative ring with an identity in which every proper homomorphic image is artinian. Cohen proved that a commutative ring R is restricted artinian if and only if it is noetherian and every nonzero prime ideal of R is maximal. Facchini and Nazemian called a commutative ring isoartinian if every descending chain of ideals becomes stationary up to isomorphism. We show that every proper homomorphic image of a commutative noetherian ring R is isoartinian if and only if R has one of the following forms: (a) R is a noetherian domain of Krull dimension one which is not a principal ideal domain; (b) R ≅ D 1 × ⋯ × D k × A 1 × ⋯ × A l {R\cong D_{1}\times\cdots\times D_{k}\times A_{1}\times\cdots\times A_{l}} , where each D i {D_{i}} is a principal ideal domain and each A i {A_{i}} is an artinian local ring (either k or l may be zero); (c) R is a noetherian ring of Krull dimension one, simple unique minimal prime ideal 𝔭 {\mathfrak{p}} , and R / 𝔭 {R/\mathfrak{p}} is a principal ideal domain. As an application of our result, we describe commutative rings whose proper homomorphic images are principal ideal rings. Some relevant examples are provided.

The stable representations of GL N over finite local principal ideal rings

Let be a discrete valuation ring with maximal ideal and with finite residue field , the field with q elements where q is a power of a prime p. For , we write for the reduction of modulo the ideal . An irreducible ordinary representation of the finite group is called stable if its restriction to the principal congruence kernel , where , consists of irreducible representations whose stabilizers modulo , where , are centralizers of certain matrices in , called stable matrices. The study of stable representations is motivated by constructions of strongly semisimple representations, introduced by Hill, which is a special case of stable representations. In this paper, we explore the construction of stable irreducible representations of the finite group for . Communicated by Scott Chapman

S-principal ideal multiplication modules

In this paper, we study S-Principal ideal multiplication modules. Let be a commutative ring with a multiplicatively closed set and an A-module. A submodule N of M is said to be an S-multiple of M if there exist and a principal ideal I of A such that . is said to be an S-principal ideal multiplication module if every submodule of is an S-multiple of M. Various examples and properties of S-principal ideal multiplication modules are given. We investigate the conditions under which the trivial extension is an -principal ideal ring. Also, we prove Cohen type theorem for S-principal ideal multiplication modules in terms of S-prime submodules.

Correction: On $ S $-principal right ideal rings

Correction: On $ S $-principal right ideal rings

A characterization of Krull domains in terms of their factor rings

ABSTRACT Let D be an integral domain, w be the so-called w-operation on D, and be the set of nonzero finitely generated ideals J of D with . Let I be a w-ideal of D, D/I be the factor ring of D modulo I, be the injective envelope of D/I, and . Recently Zhou, Kim, and Hu showed that is a commutative ring with identity containing D/I. The Krull domains are a w-analog of the Dedekind domains in the sense that D is a Krull domain if and only if every nonzero ideal of D is w-invertible. It is well known that D is a Dedekind domain if and only if D/aD is a zero-dimensional semilocal principal ideal ring (PIR) for all nonzero nonunits , if and only if D/I is a zero-dimensional semilocal PIR for all nonzero ideals I of D. In this paper, we prove that D is a Krull domain if and only if is a zero-dimensional semilocal PIR for any nonzero nonunit a of D, if and only if is a zero-dimensional semilocal PIR for any w-ideal I of D. We also study when is Noetherian, Artinian, or a (zero-dimensional) semilocal Bézout ring for all nonzero principal ideals or w-ideals I of D.

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.

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.