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In this work, we first prove a necessary and sufficient condition for a pairs of linear codes over finite rings to be linear complementary pairs (abbreviated to LCPs). In particular, a judging criterion of free LCP of codes over finite commutative rings is obtained. Using the criterion of free LCP of codes, we construct a maximum-distance-separable (MDS) LCP of codes over ring $$\mathbb {Z}_4$$ . Then, all the possible LCP of codes over chain rings are determined. We also generalize the criterions for constacyclic and quasi-cyclic LCP of codes over finite fields to constacyclic and quasi-cyclic LCP of codes over chain rings. Finally, we give a characterization of LCP of codes over principal ideal rings. Under suitable conditions, we also obtain the judging criterion for a pairs of cyclic codes over principal ideal rings $$\mathbb {Z}_{k}$$ to be LCP, and illustrate a MDS LCP of cyclic codes over the principal ideal ring $$\mathbb {Z}_{15}$$ .

Definirali bomo relacijo leve ekvivalence na matrikah, katerih elementi pripadajo nekemu glavnemu kolobarju. Poiskali bomo Hermitovo normalno formo in obravnavali njene posledice. We will define the left equivalence relation on matrices whose elements belong to a principal ideal ring. We will find the Hermite normal form and explore some of its consequences.

Let [Formula: see text] be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. In this paper we show that any finite-dimensional indecomposable [Formula: see text]-module is generated by one element. In particular, any indecomposable submodule of [Formula: see text] under the adjoint action is generated by a special element of [Formula: see text]. Using this result, we show that the Hopf algebra [Formula: see text] is a principal ideal ring, i.e., any two-sided ideal of [Formula: see text] is generated by one element. As an application, we give explicitly the generators of ideals, primitive ideals, maximal ideals and completely prime ideals of the Taft algebras.

Let Hm,n be the mn2-dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. In this paper, we show that any finite dimensional indecomposable Hm,n-module is a cy...

In this paper, we first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor...

Several different versions of “factoriality” have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the context of a commutative monoid ring R[S], determining necessary and sufficient conditions for R[S] to be various kinds of “unique factorization rings.” Our work generalizes Anderson et al.’s results about “unique factorization” in R[X], Gilmer and Parker’s characterization of factorial monoid domains, and Hardy and Shores’s classification of when R[S] is a principal ideal ring (for S cancellative). Along the way, we determine when R[S] is “restricted cancellative” or satisfies various “(restricted) ideal cancellation laws.”

In this paper we present new techniques for improving the computation of strong Grobner bases over a principal ideal ring R . More precisely, we describe how to lift a strong Grobner basis along a canonical projection R → R / n , n ≠ 0 , and along a ring isomorphism R → R 1 × R 2 . We then apply this to the computation of strong Grobner bases over a non-trivial quotient of a principal ideal domain R / n R . The idea is to run a standard Grobner basis algorithm pretending R / n R to be field. If we discover a non-invertible leading coefficient c, we use this information to try to split n = a b with coprime a , b . If this is possible, we recursively reduce the original computation to two strong Grobner bases computations over R / a R and R / b R respectively. If no such c is discovered, the returned Grobner basis is already a strong Grobner basis for the input ideal over R / n R .

Let $$(R, \pi R, k,e)$$ be a commutative special principal ideal ring (SPIR), where R is its maximal ideal, k its residual field and e the index of nilpotency of $$\pi$$ . An ideal I of R[X] is called an integral ideal if it contains a monic polynomial. In this paper, we show that if R is a SPIR, then an ideal I in R[X] is integral if and only if $${\overline{I}} \ne {{\overline{0}}}$$ in k[X]. Furthermore, the lowest degree of monic polynomials in I is exactly the lowest degree of nonzero polynomials in $${\overline{I}}$$ .

Let p be an odd prime, s and m be positive integers. Cyclic codes of length $2p^{{s}}$ over $\mathbb {F}_{{p}^{{m}}}$ are the ideals $\langle ( {x}-1)^{i}({x}+1)^{j} \rangle $ , where $0 \le {i}, {j} \le {p}^{ {s}}$ , of the principal ideal ring $\mathbb {F}_{{p}^{{m}}}[{x}]/\langle {x}^{2\textit {p}^{\textit {s}}}-1 \rangle $ . Using this structure, the symbol-pair distances of all cyclic codes of length $2\textit {p}^{\textit {s}}$ over $\mathbb F_{{p}^{{m}}}$ are completely determined. In addition, we establish all MDS symbol-pair cyclic codes of length $2\textit {p}^{\textit {s}}$ over $\mathbb F_{{p}^{{m}}}$ . Some MDS symbol-pair cyclic codes are better than all the known ones. Among others, we discuss possible applications to construct quantum MDS symbol-pair codes.

The rings considered in this article are commutative with identity. For an ideal $I$ of a ring $R$, we denote the annihilator of $I$ in $R$ by $Ann(I)$. An ideal $I$ of a ring $R$ is said to be an exact annihilating ideal if there exists a non-zero ideal $J$ of $R$ such that $Ann(I) = J$ and $Ann(J) = I$. For a ring $R$, we denote the set of all exact annihilating ideals of $R$ by $\mathbb{EA}(R)$ and $\mathbb{EA}(R)\backslash \{(0)\}$ by $\mathbb{EA}(R)^{*}$. Let $R$ be a ring such that $\mathbb{EA}(R)^{*}\neq \emptyset$. With $R$, in [Exact Annihilating-ideal graph of commutative rings, {\it J. Algebra and Related Topics} {\bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of $R$, denoted by $\mathbb{EAG}(R)$ whose vertex set is $\mathbb{EA}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent if and only if $Ann(I) = J$ and $Ann(J) = I$. In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of $\mathbb{EAG}(R)$, where either $R$ is a special principal ideal ring or $R$ is a reduced ring which admits only a finite number of minimal prime ideals.

We carry out a study of rings $R$ for which $\operatorname{Hom}_R(M,N)\neq 0$ for all nonzero $ N\leq M_R$. Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur's lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Köthe ring $R$ is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when $R$ is retractable.

Let K be the quotient field of a principal ideal ring R, and let σ = (σij) be a full (respectively, elementary) net of order n ≥ 2 (respectively, n ≥ 3) over K such that the additive subgroups σij are nonzero R-modules. It is proved that, up to conjugation by a diagonal matrix, all σij are ideals of a fixed intermediate subring P, R ⊆ P ⊆ K.

Let D 2 n = 〈 x , y ∣ x n = 1 , y 2 = 1 , y x y = x − 1 〉 be a dihedral group, and R = GR ( p 2 , m ) be a Galois ring of characteristic p 2 and cardinality p 2 m where p is a prime. Left ideals of the group ring R [ D 2 n ] are called left dihedral codes over R of length 2 n , and abbreviated as left D 2 n -codes over R . Let gcd ( n , p ) = 1 in this paper. Then any left D 2 n -code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A i and outer codes C i , where A i is a cyclic code over R of length n and C i is a skew cyclic code of length 2 over a Galois ring or principal ideal ring extension of R . Specifically, a generator matrix and basic parameters for each outer code C i are given. A formula to count the number of these codes is obtained and the dual code for each left D 2 n -code is determined. Moreover, all self-dual left D 2 n -codes and self-orthogonal left D 2 n -codes over R are presented.

Because of its interesting applications in coding theory, cryptography, and algebraic combinatorics, in recent decades a lot of attention has been paid to the algebraic structure of the ring of polynomials R[x], where R is a finite commutative ring with identity. Motivated by this popularity, in this paper we determine when R[x] is a principal ideal ring. In fact, we prove that R[x] is a principal ideal ring if and only if R is a finite direct product of finite fields. MSC2010: 13F10, 13F20, 16P10, 13C05. Resumen Debido a sus interesantes aplicaciones en teoría de códigos, criptografía y combinatoria algebraica, en décadas recientes se ha incrementado la atención en la estructura algebraica del anillo de polinomios R[x], donde R es un anillo conmutativo finito con identidad. Motivados por esta popularidad, en este artículo determinamos cuándo R[x] es un anillo de ideales principales. De hecho, demostramos que R[x] es un anillo de ideales principales, si y sólo si, R es un producto directo finito de campos finitos.

Let R be a (possibly noncommutative) finite principal ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module $$R^n$$ is produced. This is used to set up a greedy algorithm that selects vectors for which all linear combinations with the previously selected vectors satisfy a pre-specified selection property and updates the to-be-constructed code to the linear hull of the vectors selected so far. The output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. In this paper we investigate the properties of such lexicodes over finite principal ideal rings and show that the total ordering of the ring elements has to respect containment of ideals for the algorithm to produce meaningful results. Only then it is guaranteed that the algorithm is exhaustive and thus produces codes that are maximal with respect to inclusion. It is further illustrated that the output of the algorithm heavily depends on the total ordering and chosen basis.

In this work, we study the structure of multivariable modular codes over finite chain rings when the ambient space is a principal ideal ring. We also provide some applications to additive modular codes over the finite field F 4 .

In this paper, we consider the ideal based zero divisor graph ΓI(R) of a commutative ring R. We discuss some graph theoretical properties of ΓI(R) in relation with zero divisor graph. We also relate certain parameters like vertex chromatic number, maximum degree and minimum degree for the graph ΓI(R) with that of Γ(R/I ). Further we determine a necessary and sufficient condition for the graph to be Eulerian and regular.

We study the indecomposability and isomorphism of objects from the category of monomial matrices Mmat(K) over a commutative local principal ideal ring K (whose objects are square monomial matrices and the morphisms from X to Y are matrices C such that XC = CY). We also study the subcategory Mmat0(K) of the category Mmat(K) with the same objects and solely with morphisms that are monomial matrices.

In this paper, the determinants of n × n matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of n × n matrices over a commutative finite chain ring R of a fixed determinant a is determined for all a ∈ R and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of n × n matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.

The Nagata ring R(X) and the Serre’s conjecture ring R⟨X⟩ are two localizations of the polynomial ring R[X] at the polynomials of unit content and at the monic polynomials, respectively. In this pa...

The study of the class of van der Waerden rings was initiated by Comfort, Remus and Szambien in [7] . A compact ring ( R , T ) is a van der Waerden ring if and only if T is the only totally bounded ring topology on R . In this paper the study of this class of compact rings is continued. We prove that an Abelian compact van der Waerden ring R has the form R = ( ∏ i ∈ I R i ) × L , where L is a compact radical ring with radical topology and R i ( i ∈ I ) is a compact local topologically finitely generated ring. It is shown that every van der Waerden ring is totally disconnected and metrizable. Furthermore, the class of van der Waerden rings is closed under extensions. Different classes of compact rings are studied which are closely related to van der Waerden rings.

An interesting result, obtaining by some theorems of Asano, Kothe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ℳ of the form ℳ = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (res...

We prove that for any valuation ring R of Krull dimension ≤1 or any commutative local ring R with Krull dimension 0 and n≥3, the special linear group SLn(R[x]) coincides with the group of elementary matrices En(R[x]), and for an arbitrary arithmetical ring R of Krull dimension ≤1 and n≥3, the group SLn(R[x])=SLn(R)⋅En(R[x]). These give partial solutions to two conjectures of Yengui.

Let R be a ring and $$\delta $$ is a derivation of R. In this paper, it is proved that, under suitable conditions, the differential polynomial ring $$R[x;\delta ]$$ has the same triangulating dimension as R. Furthermore, for a piecewise prime ring, we determine a large class of the differential polynomial ring which have a generalized triangular matrix representation for which the diagonal rings are prime.

We prove that a ring which is a sum of two PI subrings is a PI ring as well. This answers a question of K. I. Beidar and A. V. Mikhalev that has been open since 1995.

Let R be a semiprime ring with center Z(R) and with extended centroid C. Suppose that τ:R→R is an anti-homomorphism such that the image of τ has small centralizer. It is proved that the fol...

Let R be a commutative ring with identity and let Nil ( R ) be the ideal of all nilpotent elements of R . Let I ( R ) = { I : I is a non-trivial ideal of R and there exists a non-trivial ideal J such that I J ⊆ Nil ( R ) } . The nil-graph of ideals of R is defined as the simple undirected graph A G N ( R ) whose vertex set is I ( R ) and two distinct vertices I and J are adjacent if and only if I J ⊆ Nil ( R ) . In this paper, we study the planarity and genus of A G N ( R ) . In particular, we have characterized all commutative Artin rings R for which the genus of A G N ( R ) is either zero or one.

In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation.

Let R be a finite commutative ring with identity. In this paper, given an integer $$k\ge 2$$ , we obtain an exact formula for the number of ways to represent each element of R as the sum of k exceptional units. This generalizes a recent result of Quan-Hui Yang and Qing-Qing Zhao for the case where R is the ring $${\mathbb {Z}}_n$$ of residue classes modulo n.