Principal Ideal Ring Research Articles

Matrix graphs and MRD codes over finite principal ideal rings

Let R be a finite principal ideal ring and m,n,d positive integers. In this paper, we study the matrix graph over R which is the graph whose vertices are m×n matrices over R and two matrices A and B are adjacent if and only if 0<rank(A−B)<d. We show that this graph is a connected vertex transitive graph. The distance, diameter, independence number, clique number and chromatic number of this graph are also determined. This graph can be applied to study MRD codes over R. We obtain that a maximal independent set of the matrix graph is a maximum rank distance (MRD) code and vice versa. Moreover, we show the existence of linear MRD codes over R.

Categorial properties of compressed zero-divisor graphs of finite commutative rings

By modifying the existing definition of a compressed zero-divisor graph [Formula: see text], we define a compressed zero-divisor graph [Formula: see text] of a finite commutative unital ring [Formula: see text], where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of [Formula: see text]). We prove that this is the best possible compression which induces a functor [Formula: see text], and that this functor preserves categorial products (in both directions). We use the structure of [Formula: see text] to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.

Homogeneous metric and matrix product codes over finite commutative principal ideal rings

In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of such codes are also explicitly investigated.

$G$-codes over formal power series rings and finite chain rings

In this work, we define $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. We show that the dual of a $G$-code is again a $G$-code in this setting. We study the projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing $\gamma$-adic codes over $R_\infty$ to $\gamma$-adic $G$-codes over the same ring. We also study $G$-codes over principal ideal rings.

On projective intersection graph of ideals of commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] the set of all nontrivial proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as the set [Formula: see text], and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring [Formula: see text], [Formula: see text] is a projective graph if and only if [Formula: see text] where [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] of nilpotency three and [Formula: see text] is a field. Furthermore, it is shown that for an Artinian ring [Formula: see text] [Formula: see text] if and only if [Formula: see text] where each [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] such that [Formula: see text]

Rank-Metric Codes Over Finite Principal Ideal Rings and Applications

In this paper, it is shown that some results in the theory of rank-metric codes over finite fields can be extended to finite commutative principal ideal rings. More precisely, the rank metric is generalized and the rank-metric Singleton bound is established. The definition of Gabidulin codes is extended and it is shown that its properties are preserved. The theory of Gröbner bases is used to give the unique decoding, minimal list decoding, and error-erasure decoding algorithms of interleaved Gabidulin codes. These results are then applied in space-time codes and in random linear network coding as in the case of finite fields. Specifically, two existing encoding schemes of random linear network coding are combined to improve the error correction.

Efficient Gröbner bases computation over principal ideal rings

In this paper we present new techniques for improving the computation of strong Gröbner bases over a principal ideal ring R. More precisely, we describe how to lift a strong Gröbner basis along a canonical projection R→R/n, n≠0, and along a ring isomorphism R→R1×R2. We then apply this to the computation of strong Gröbner bases over a non-trivial quotient of a principal ideal domain R/nR. The idea is to run a standard Gröbner basis algorithm pretending R/nR to be field. If we discover a non-invertible leading coefficient c, we use this information to try to split n=ab with coprime a,b. If this is possible, we recursively reduce the original computation to two strong Gröbner bases computations over R/aR and R/bR respectively. If no such c is discovered, the returned Gröbner basis is already a strong Gröbner basis for the input ideal over R/nR.

When is every module with essential socle a direct sum of automorphism-invariant modules?

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

Another type of dimensions of modules and rings

We say that a class Q of left R-modules is a monic class if a nonzero submodule of a module in Q is also a module in Q. For a monic class Q, we define a Q-dimension of modules that measures how far modules are from the modules in Q. For a monic class Q of indecomposable modules we characterize rings whose modules have Q-dimension. We prove that for an artinian principal ideal ring the Q-dimension coincides with the uniserial dimension. We also characterize when every module has Q-dimension.

Rational Recursion Operators for Integrable Differential\u2013Difference Equations

In this paper we introduce the concept of preHamiltonian pairs of difference operators, demonstrate their connections with Nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential–difference equations. We begin with a rigorous setup of the problem in terms of the skew field of rational (pseudo–difference) operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational (pseudo–difference) operators. In particular, we give a criteria for a rational operator to be weakly nonlocal. A difference operator is called preHamiltonian, if its image is a Lie subalgebra with respect to the Lie bracket on the difference field. Two preHamiltonian operators form a preHamiltonian pair if any linear combination of them is preHamiltonian. Then we show that a preHamiltonian pair naturally leads to a Nijenhuis operator, and a Nijenhuis operator can be represented in terms of a preHamiltonian pair. This provides a systematic method to check whether a rational operator is Nijenhuis. As an application, we construct a preHamiltonian pair and thus a Nijenhuis recursion operator for the differential–difference equation recently discovered by Adler and Postnikov. The Nijenhuis operator obtained is not weakly nonlocal. We prove that it generates an infinite hierarchy of local commuting symmetries. We also illustrate our theory on the well known examples including the Toda, the Ablowitz–Ladik, and the Kaup–Newell differential–difference equations.

Linear complementary dual codes over rings

By using linear algebra over finite commutative rings, we will present some judging criterions for linear complementary dual (LCD) codes over rings, in particular, free LCD codes over finite commutative rings are described. By using free LCD codes over finite commutative rings and the Chinese Remainder Theorem, LCD codes over semi-simple rings are constructed and the equivalence of free codes and free LCD codes is given. In addition, all the possible LCD codes over chain rings are determined. We also generalize the judging criterion for cyclic LCD codes over finite fields to cyclic LCD codes over chain rings. Based on the above results and the Chinese Remainder Theorem, we also present results for LCD codes over principal ideal rings.

On a Class of Constacyclic Codes of Length 4ps over Fpm[u]〈 ua 〉

For any odd prime p such that pm ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring [Formula: see text] that have the form Λ = Λ0 + uΛ1 + ⋯ + ua−1 Λa−1, where Λ0, Λ1, …, Λa−1 ∊ 𝔽pm, Λ0 ≠ 0, Λ1 ≠ 0. The class of Λ-constacyclic codes of length 4ps over ℛa is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4ps is expressed as a direct sum of a −λ-constacyclic code and a λ-constacyclic code of length 2ps. In the main case that the unit Λ is not a square, we prove that the polynomial x4 − λ0 can be decomposed as a product of two quadratic irreducible and monic coprime factors, where [Formula: see text]. From this, the ambient ring [Formula: see text] is proven to be a principal ideal ring, whose maximal ideals are ⟨x2 + 2ηx + 2η2⟩ and ⟨x2 − 2ηx + 2η2⟩, where λ0 = −4η4. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4ps over ℛa. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.

Characterization of multiplication commutative rings with finitely many minimal prime ideals

The aim of the article is to give a characterization of a multiplication commutative ring with finitely many minimal prime ideals: Each such ring is a finite direct product of rings where Di is either a Dedekind domain or an Artinian, local, principal ideal ring and vice versa. In particular, each such ring is a Noetherian ring. As a corollary subclasses of such rings are described (semiprime, Artinian, semiprime and Artinian, local, domain, etc.).

Weil representations of unitary groups over ramified extensions of finite local rings with odd nilpotency length

We find the irreducible decomposition of the Weil representation of the unitary group , where A is a ramified quadratic extension of a finite, commutative, local, principal ideal ring R and the nilpotency degree of the maximal ideal of A is odd. We show in particular that this Weil representation is multiplicity free. Restriction to the special unitary group preserves irreducibility and multiplicity freeness provided n > 1.

Prime virtually semisimple modules and rings

This article is a sequel to the recent three papers on “virtually semisimple modules and rings,” by Behboodi et al., which two of them appeared in the Algebras and Representation Theory and Communications in Algebra in 2018. An R-module M is called virtually semisimple if each submodule of M is isomorphic to a direct summand. A ring R is called left (resp., right) virtually semisimple if (resp., RR) is virtually semisimple. In this article, we study rings and modules in which every prime submodule is isomorphic to a direct summand, and called them prime virtually (or -virtually) semisimple modules. A ring R is called left (resp., right) -virtually semisimple if (resp., RR) is -virtually semisimple. The results of the article are inspired by a characterization of left -virtually semisimple rings. We prove that these rings are precisely the left virtually semisimple rings, and in this case , where each Di is a domain and each is a principal left ideal ring. We also answer to the following questions: (i) Describe rings R where each (finitely generated or cyclic) left R-module is -virtually semisimple?, and (ii) Describe rings R where each left R-module is a direct sum of indecomposable -virtually semisimple modules? Finally, we study -virtually semisimple modules over commutative rings.

Parainjectivity, paraprojectivity and artinian principal ideal rings

In this work, we introduce the concepts of parainjectivity and paraprojectivity. We give some basic properties about them and we obtain some characterizations of artinian principal ideal rings.

Random Motion on Finite Rings, I: Commutative Rings

We consider irreversible Markov chains on finite commutative rings randomly generated using both addition and multiplication. We restrict ourselves to the case where the addition is uniformly random and multiplication is arbitrary. We first prove formulas for eigenvalues and multiplicities of the transition matrices of these chains using the character theory of finite abelian groups. The examples of principal ideal rings (such as $\mathbb{Z}_{n}$) and finite chain rings (such as $\mathbb{Z}_{p^k}$) are particularly illuminating and are treated separately. We then prove a recursive formula for the stationary probabilities for any ring, and use it to prove explicit formulas for the probabilities for finite chain rings when multiplication is also uniformly random. Finally, we prove constant mixing time for our chains using coupling.

On (α + uβ)-constacyclic codes of length 4ps over 𝔽pm + u𝔽pm∗

For any odd prime [Formula: see text] such that [Formula: see text], the structures of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text] [Formula: see text] are established in term of their generator polynomials. When the unit [Formula: see text] is a square, each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of two constacyclic codes of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that the ambient ring [Formula: see text] is a principal ideal ring. From that, the structure, number of codewords, duals of all such [Formula: see text]-constacyclic codes are obtained. As an application, we identify all self-orthogonal, dual-containing, and the unique self-dual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text].

On commutative rings whose ideals are direct sums of cyclic modules

A generalization of Köthe rings is the family of rings whose ideals are direct sums of cyclic modules. These rings were previously studied in the commutative local case. This motivated us to study commutative rings with local dimension whose ideals are direct sums of cyclic modules. First, we obtain an structure theorem for rings of local dimension [Formula: see text]. Then we conclude that, for a ring of local dimension [Formula: see text], every ideal of [Formula: see text] is a direct sum of cyclic modules if and only if every indecomposable ideal of [Formula: see text] is cyclic if and only if every maximal ideal of [Formula: see text] is cyclic if and only if every maximal ideal of [Formula: see text] is cyclic if and only if [Formula: see text] is a principal ideal ring.

A minimal ring extension of a large finite local prime ring is probably ramified

Given any minimal ring extension [Formula: see text] of finite fields, several families of examples are constructed of a finite local (commutative unital) ring [Formula: see text] which is not a field, with a (necessarily finite) inert (minimal ring) extension [Formula: see text] (so that [Formula: see text] is a separable [Formula: see text]-algebra), such that [Formula: see text] is not a Galois extension and the residue field of [Formula: see text] (respectively, [Formula: see text]) is [Formula: see text] (respectively, [Formula: see text]). These results refute an assertion of G. Ganske and McDonald stating that if [Formula: see text] are finite local rings such that [Formula: see text] is a separable [Formula: see text]-algebra, then [Formula: see text] is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let [Formula: see text] be a finite special principal ideal ring (SPIR), but not a field, such that [Formula: see text] has index of nilpotency [Formula: see text] ([Formula: see text]). Impose the uniform distribution on the (finite) set of ([Formula: see text]-algebra) isomorphism classes of the minimal ring extensions of [Formula: see text]. If [Formula: see text] (for instance, if [Formula: see text]), the probability that a random isomorphism class consists of ramified extensions of [Formula: see text] is at least [Formula: see text]; if [Formula: see text] (for instance, if [Formula: see text] for some odd prime [Formula: see text]), the corresponding probability is at least [Formula: see text]. Additional applications, examples and historical remarks are given.