FINITE PRINCIPAL IDEAL RINGS

We give a negative answer to a question posed by A. V. Jategaonkar: is it not true that an arbitrary primary principal left ideal ring is a factor of a prime principal left ideal ring? We give a counter example in the class of finite complete primary principal ideal rings, the so-called Galois-Eisenstein-Ore rings.

Circulant matrices form an important class of matrices that have been continuously studied due to their nice algebraic structures and wide applications. In this study, we focus specifically on negacirculant matrices, which are known as extensions of the classical circulant matrices. The algebraic structures of the rings of n × n n\times n negacirculant matrices over finite fields and over finite commutative chain rings are presented. Subsequently, the algebraic structures and enumeration of the unit groups of such matrix rings are established. Additionally, the number of non-singular n × n n\times n negacirculant matrices with prescribed determinant is given in some cases. Conjectures and open problems are proposed as well as a brief discussion in the case where the underlying ring is a finite commutative ring with identity is also presented.

Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or else a lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. Lattice isomorphisms of finite one-generated rings with identity are studied. We elucidate the general structure of finite one-generated rings with identity and also give necessary and sufficient conditions for a finite ring decomposable into a direct sum of Galois rings to be generated by one element. Conditions are found under which the projective image of a ring decomposable into a direct sum of finite fields is a one-generated ring. We look at lattice isomorphisms of one-generated rings decomposable into direct sums of Galois rings of different types. Three main types of Galois rings are distinguished: finite fields, rings generated by idempotents, and rings of the form GR(pn,m), where m > 1 and n > 1. We specify sufficient conditions for the projective image of a one-generated ring decomposable into a sum of Galois rings and a nil ideal to be generated by one element.

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p e , l) (including $${\mathbb{Z}_{p^e}}$$ ). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 e , l) of length n = 2 l for any a ? 1 and l ? 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.

For an element $x$ of a commutative ring $R$, let $\Gamma_x(R)$ be the graph whose vertices are the elements of $R$ that divide $x$ such that distinct vertices $r$ and $s$ are adjacent if and only if $rs=x$. If $x$ is a nonzero nonirreducible nonunit of an integral domain $R$, then $\Gamma_x^\mathcal{C}(R)$ is the graph whose vertices are the associate classes of divisors of $x$ that are neither units nor associates of $x$ such that distinct vertices $A$ and $B$ are adjacent if and only if $rs$ divides $x$ for some (and hence every) $r\in A$ and $s\in B$. The graphs $\Gamma_x(R)$ are considered when $R$ is a local Artinian principal ideal ring, and $\Gamma_x^\mathcal{C}(R)$ is examined when $R$ is a valuation domain. For example, it is shown that a finite local ring $R$ is a principal ideal ring if and only if there exists $x\in R$ such that every connected component of $\Gamma_x(R)$ is a star graph of a prescribed cardinality. Moreover, it is proved that an integral domain $R$ is a discrete valuation ring if and only if its collection of graphs $\Gamma_x^\mathcal{C}(R)$ consists precisely of a single-vertex graph, along with every (up to isomorphism) graph that is realizable as the compressed $0$-divisor graph of a local Artinian principal ideal ring. Certain graphs associated with partially ordered abelian groups have an essential role in the work.

It is very well known that algebraic structures have valuable applications in the theory of error-correcting codes. Blake [Codes over certain rings, Inform. and Control 20 (1972) 396–404] has constructed cyclic codes over ℤm and in [Codes over integer residue rings, Inform. and Control 29 (1975), 295–300] derived parity check-matrices for these codes. In [Linear codes over finite rings, Tend. Math. Appl. Comput. 6(2) (2005) 207–217]. Andrade and Palazzo present a construction technique of cyclic, BCH, alternant, Goppa and Srivastava codes over a local finite ring B. However, in [Encoding through generalized polynomial codes, Comput. Appl. Math. 30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text] and encoding, Comput. Math. Appl. 62 (2011) 1645–1654], Shah et al. extend this technique of constructing linear codes over a finite local ring B via monoid rings [Formula: see text], where p = 2 and k = 1, 2, respectively, instead of the polynomial ring B[X]. In this paper, we construct these codes through the monoid ring [Formula: see text], where p = 2 and k = 1, 2, 3. Moreover, we also strengthen and generalize the results of [Encoding through generalized polynomial codes, Comput. Appl. Math.30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text]] and [Encoding, Comput. Math. Appl.62 (2011) 1645–1654] to the case of k = 3.

In this paper we show that finite commutative rings with unity generated by an invertible element can be characterized as certain direct sums of Galois rings. This result is used to analyze the structure of rings which support discrete Fourier transforms.

Let R be an arbitrary commutative finite chain ring and \(\gamma \) a fixed generator of the maximal ideal of R. Suppose s is the nilpotency index of \(\gamma \) and F is the residue field of R modulo its ideal \(\gamma R\), i.e. \(F = R / \gamma R\), \(\vert F \vert = q\) with \(q=p^{\alpha }\) for some prime number p and let \(R^{\times }\) denote the multiplicative group of units of R. For any \(\omega \in R^{\times }\) and \(t \ge \lceil \frac{s}{2}\rceil \), the structural properties and dual codes of \((1+ \omega \gamma ^{t})\)-quasi-twisted (QT) codes of length \(n=\ell m\), with \((m, p)=1\), over R are given. The key idea is to view a \((1 + \omega \gamma ^{t})\)-QT code over R as a linear code over \(R_{m} = R[x] / \langle x^{m} - (1 + \omega \gamma ^{t}) \rangle \). Furthermore, given the decomposition of a \((1+ \omega \gamma ^{t})\)-QT code, we provide the decomposition of its dual code. As a result, a characterization of self-dual \((1+ \omega \gamma ^{t})\)-QT codes over a finite chain ring R, with \((1+ \omega \gamma ^{t})=(1+ \omega \gamma ^{t})^{-1}\), is provided. By using the Chinese remainder theorem or the discrete Fourier transform, the ring \(R[x] / \langle x^{m} - (1 + \omega \gamma ^{t}) \rangle \) can be decomposed into a direct sum of finite chain rings. The inverse transform of the discrete Fourier transform produces a method for deriving \((1+ \omega \gamma ^{t})\)-QT codes from codes of lower lengths.

Properties of a new complex number-theoretic z-transform (CNT z-transform) over a finite ring are presented here and related to the usual z-transform. Using the Chinese remainder theorem, it is convenient to use finite rings that are isomorphic to the direct sum of finite or Galois fields of the form GF(q2) where q is a Mersenne prime. Many properties of the usual z-transform are preserved in the CNT z-transform. This transform is used in the present paper to design both recursive and nonrecursive FIR filters on a finite ring. The advantages of the FIR filter on a finite ring are the following: 1) the absence of a roundoff error build up in the computation of either the recursive or nonrecursive realization of the filter; 2) when the FIR filter is recursive, the question of stability does not arise as long as the magnitudes of the impulse response and the input sequence do not exceed their design values; 3) for the frequency sample representation of the FIR filter an absolute error bound on the impulse response function can be obtained in terms of the power spectrum. The time required to compute a nonrecursive FIR filter on the Galois field GF(q2), where q is a Mersenne prime, is competitive with the similar nonrecursive realization on the usual complex number field, using the FFT algorithm.

On some Euclidean properties of matrix algebras

In this paper, we study constacyclic codes over finite principal ideal rings. An isomorphism between constacyclic codes and cyclic codes over finite principal ideal rings is given. Further, an open question is partially answered by giving necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite principal ideal rings. As an example of codes over a finite principal ideal ring, we study constacyclic codes over R+vR where v2=v and R is a finite chain ring.

We prove that if $R$ is a principal ideal ring and $A\in \mathrm {M}_{n}(R)$ is a matrix with trace zero, then $A$ is a commutator, that is, $A=XY-YX$ for some $X,Y\in \mathrm {M}_{n}(R)$. This generalises the corresponding result over fields due to Albert and Muckenhoupt, as well as that over $\mathbb {Z}$ due to Laffey and Reams, and as a by-product we obtain new simplified proofs of these results. We also establish a normal form for similarity classes of matrices over PIDs, generalising a result of Laffey and Reams. This normal form is a main ingredient in the proof of the result on commutators.

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.

Let R be a local Artin principal ideal ring, R[x] the polynomial ring over R with indeterminate x. Let π be an element of R such that is the unique maximal ideal of R. Let I be a zero-dimensional ideal of R[x], and \(\) the radical ideal of I. In this paper we show that I is the annihilating ideal of a linear recurring sequence over R if and only if I satisfies the following formula $$$$

The study of finite extension of Galois rings in the recent past have given rise to commutative completely primary finite rings that have attracted much attention as they have yielded important results towards classification of finite rings into well-known structures. In this paper, we give a construction of a class of completely primary finite ringof characteristicwhose subsets of zero divisors satisfy the condition . The ring is constructed over its subring as an idealization of the - modules. A thorough determination and classification of the structure of the group of invertible elements using fundamental theorem of finitely generated abelian groups is given.