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In this article, based on the defining set $$D=\{x\in \mathbb {F}_{p^m}^*:Tr(x)=0\}$$ , we explore the Lee-weight distribution of linear codes $${\mathcal {C}}_D=\{(tr(ax^2))_{x\in D}:a\in \mathbb {F}_{p^m}+u\mathbb {F}_{p^m}\}$$ over the finite ring $$\mathbb {F}_p+u\mathbb {F}_p$$ with p being an odd prime and $$u^2=0$$ . By employing the exponential and Gauss sums, we calculate the Lee weight of all possible codewords as well as their frequencies. Two classes of eight-weight linear codes are obtained, where one of them is new. We also show that for some small values m, the code $${\mathcal {C}}_D$$ has two weights ( $$m=2$$ ) and seven weights ( $$m=3,4$$ and $$p=3$$ ).

In this paper, first we consider the linear complexity of quaternary sequences over the finite ring of order four and the finite field of order four. These sequences are constructed from new generalized cyclotomic classes modulo pn. Second, we study the linear complexity of new generalized cyclotomic binary sequences of period 2pn recently proposed by Ouyang et al. (Des. Codes Cryptogr. 87(5), 1–12 2019). We generalized results presented in this work and discuss the author’s conjecture. In conclusion, we again derive the linear complexity of quaternary sequences with period 2pn over the finite ring of order four and the finite field of order four. We use generalized cyclotomic classes modulo 2pn to define these sequences.

The extension problem underlies notions of code equivalence. Two approaches to the extension problem are described. One is a matrix approach that reduces the general problem for weights to one for symmetrized weight compositions. The other is a monoid algebra approach that reframes the extension problem in terms of modules over the monoid algebra determined by the multiplicative monoid of a finite ring.

WOS: 000531286600021 The DNA 4-bases are matched with the 256 elements of the finite ring R=F4+uF4+vF4+uvFR=F_{4}+uF_{4}+vF_{4}+uvF_{4}$$\end{document}, where u2=u,v2=v,uv=vu. By defining a non trivial automorphism over R, the skew cyclic codes over the finite ring R are introduced. The reversible DNA codes are obtained, by using the skew cyclic codes whose generator polynomials satisfy some conditions. Thanks to this, the reversibility problem for DNA 4-bases can be solved. Moreover, the Gray images of the skew cyclic codes over the finite ring are determined.

For a ring R, let J(R) and C(R) be the Jacobson radical and the center of R, respectively. It is proved that a left uniserial, centrally essential finite ring is commutative. In the proof of this result, we use the following more general assertion: if F is a field, then any left uniserial, left Artinian, centrally essential ring R with factor field $$R/J(R)\cong F$$ is commutative if and only if the field F does not have non-zero derivations. It is proved that there are non-commutative uniserial Artinian centrally essential rings and right uniserial, right Noetherian PI rings which are not centrally essential. It is also proved that a left Artinian, left uniserial ring R with J(R) of nilpotence index n is centrally essential provided $$\displaystyle {J(R)^{\left[ n/2\right] }\subseteq C(R)}$$.

We prove a formula for the number of representations of an element in a finite basic ring as a sum of k exceptional units and find bounds for this number in an arbitrary finite ring with identity.

We present a new and simpler proof of a combinatorial nature for the commutative case of Wood’s theorem which states that a finite ring has a generating character if and only if it is a Frobenius ring.

Let D be a subset of a finite commutative ring R with identity. Let $$f(x)\in R[x]$$ be a polynomial of degree d. For a nonnegative integer k, we study the number $$N_f(D,k,b)$$ of k-subsets S in D such that $$\begin{aligned} \sum _{x\in S} f(x)=b. \end{aligned}$$ In this paper, we establish several bounds for the difference between $$N_f(D,k, b)$$ and the expected main term $$\frac{1}{|R|}{|D|\atopwithdelims ()k}$$ , depending on the nature of the finite ring R and f. For $$R=\mathbb {Z}_n$$ , let $$p=p(n)$$ be the smallest prime divisor of n, $$|D|=n-c \ge C_dn p^{-\frac{1}{d}}\,+\,c$$ and $$f(x)=a_dx^d +\cdots +a_0\in \mathbb {Z}[x]$$ with $$(a_d, \ldots , a_1, n)=1$$ . Then $$\begin{aligned} \left| N_f(D, k, b)-\frac{1}{n}{n-c \atopwithdelims ()k}\right| \le {\delta (n)(n-c)+(1-\delta (n))\left( C_dnp^{-\frac{1}{d}}+c\right) +k-1\atopwithdelims ()k}, \end{aligned}$$ answering an open question raised by Stanley (Enumerative combinatorics, 1997) in a general setting, where $$\delta (n)=\sum _{i\mid n, \mu (i)=-1}\frac{1}{i}$$ and $$C_d=e^{1.85d}$$ . Furthermore, if n is a prime power, then $$\delta (n) =1/p$$ and one can take $$C_d=4.41$$ . Similar and stronger bounds are given for two more cases. The first one is when $$R=\mathbb {F}_q$$ , a q-element finite field of characteristic p and f(x) is general. The second one is essentially the well-known subset sum problem over an arbitrary finite abelian group. These bounds extend several previous results.

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.

In this paper, we establish a mass formula for self-dual codes over the finite chain ring $$\mathbb {F}_q+u\mathbb {F}_q+u^2\mathbb {F}_q$$ , where $$\mathbb {F}_q$$ is the finite field of order q and $$u^3=0$$ . We also give a classification of self-dual codes over $$\mathbb {F}_q+u\mathbb {F}_q+u^2\mathbb {F}_q$$ , for $$q=2,3,4,5,7,8,9$$ , with lengths 2 and 4.

Let R be a commutative ring with $$1\ne 0$$ and the additive group $$R^+$$ . Several graphs on R have been introduced by many authors, among zero-divisor graph $$\Gamma _1(R)$$ , co-maximal graph $$\Gamma _2(R)$$ , annihilator graph AG(R), total graph $$ T(\Gamma (R))$$ , cozero-divisors graph $$\Gamma _\mathrm{c}(R)$$ , equivalence classes graph $$\Gamma _\mathrm{E}(R)$$ and the Cayley graph $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when R is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if R has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results, we prove that for a commutative finite ring R with $$|\mathrm{Max}(R)|=n \ge 3$$ , $$ \Gamma _1(R) \simeq \Gamma _2(R)$$ if and only if $$R\simeq \mathbb {Z}^n_2$$ ; if and only if $$\Gamma _1(R) \simeq \Gamma _\mathrm{E}(R)$$ . Also the annihilator graph is identical to the cozero-divisor graph if and only if R is a Frobenius ring.

For each non-commutative ring R, the commuting graph of R is a graph with vertex set $R\setminus Z(R)$ and two vertices $x$ and $y$ are adjacent if and only if $x\neq y$ and $xy=yx$. In this paper, we consider the domination and signed domination numbers on commuting graph $\Gamma(R)$ for non-commutative ring $R$ with $Z(R)=\{0\}$. For a finite ring $R$, it is shown that $\gamma(\Gamma(R)) + \gamma(\overline{\Gamma}(R))=|R|$ if and only if $R$ is non-commutative ring on 4 elements. Also we determine the domination number of $\Gamma(\prod_{i=1}^{t}R_i)$ and commuting graph of non-commutative ring $R$ of order $p^3$, where $p$ is prime. Moreover we present an upper bound for signed domination number of $\Gamma(\prod_{i=1}^{t}R_i)$., Comment: 9 pages in Iranian Journal of Science and Technology, series A- first online 13 June 2016

We first consider the group inverses of the block matrices $$\left( {\begin{array}{*{20}{c}} A&B \\ 0&C \end{array}} \right)$$ over a weakly finite ring. Then we study the sufficient and necessary conditions for the existence and the representations of the group inverses of the block matrices $$\left( {\begin{array}{*{20}{c}} A&B \\ C&D \end{array}} \right)$$ over a ring with unity 1 under the following conditions respectively: (i) B = C, D = 0, B # and (B π A)# both exist; (ii) B is invertible and m = n; (iii) A # and (D − CA # B)# both exist, C = CAA #, where A and D are m × m and n × n matrices, respectively.

In this paper, we give examples of infinite series of finite rings B () , where m ≥ 2, 0 ≤ v ≤ p−1, and p is a prime number, that are not representable by matrix rings over commutative rings, and we describe the basis of polynomial identities of these rings. We prove here that every variety var B () , where m = 2 or m − 1 = (p − 1)k, k ≥ 1, and p ≥ 3 or p = 2 and m ≥ 3, 0 ≤ v < p, and p is a prime number, is a minimal variety containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring. Therefore, we describe almost finitely representable varieties of rings whose generating ring contains an idempotent element of additive order p.

Tang et al. and Lim et al. presented ways to construct balanced quaternary sequences with even period and optimal autocorrelation value by inverse Gray-mapping of binary sequences with optimal autocorrelation value. In this article, we consider quaternary sequences constructed from binary Legendre or Hall's sextic sequence by these methods. We derive the linear complexity of series of balanced quaternary sequences with optimal autocorrelation value over the finite ring of four elements.

We give a complete description of subdirectly irreducible finite associative rings with commuting nilpotent elements. Also it is proved that a finite ring the nilpotent elements of which commute is representable by matrices over a commutative ring.

The zero-divisor graph of an associative ring R is a graph such that its vertices are all nonzero (one-sided and two-sided) zero-divisors, and moreover, two distinct vertices x and y are joined by an edge iff xy = 0 or yx = 0. We give a complete description of varieties of associative rings in which all finite rings have Hamiltonian zero-divisor graphs. Also finite decomposable rings with unity having Hamiltonian zero-divisor graphs are characterized.

We describe all possible ways how a ring can be expressed as the union of three of its proper subrings. This is an analogue for rings of a 1926 theorem of Scorza about groups. We then determine the minimal number of proper subrings of the simple matrix ring $M_{n}(q)$ whose union is $M_{n}(q)$.

In [Jain, S.: Array codes in the generalized-Lee-RT-pseudo-metric (the GLRTP-metric), to appear in Algebra Colloq.], Jain introduced a new pseudo-metric on the space Matm × s(Zq), the module space of all m × s matrices with entries from the finite ring Zq, generalized the classical Lee metric [Lee, C. Y.: Some properties of non-binary error correcting codes. IEEE Trans. Inform. Theory, IT-4, 77–82 (1958)] and array RT-metric [Rosenbloom, M. Y., Tsfasman, M. A.: Codes for m-metric. Prob. Inf. Transm., 33, 45–52 (1997)] and named this pseudo-metric as the Generalized-Lee-RT-Pseudo-Metric (or the GLRTP-Metric). In this paper, we obtain some lower bounds for two-dimensional array codes correcting CT burst array errors [Jain, S.: CT bursts — from classical to array coding. Discrete Math., 308–309, 1489–1499 (2008)] with weight constraints under the GLRTP-metric.

A general scheme is investigated that is destined for obtaining estimates based on the cardinality of subsets of a fixed set of automata over some finite commutative-associative ring with unit element. A scheme is proposed to solve parametric systems of polynomial equations based on classes of associated elements of the ring. Some general characteristics of automata over the ring are established.

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.

The structure of a class of automata is analyzed. These automata are analogs of symmetric chaotic dynamical systems over a finite ring, namely, the Guckenheimer---Holmes cycle and free-running systems. Problems of parametric identification and identification of initial states are solved, and a set of fixed points of automaton mappings is characterized.

A class of homogenous systems of random nonlinear equations over an arbitrary finite ring with left unity is considered. The author analyzes the invariance boundaries for limit factorial moments of nonzero solutions, the limit distribution of the number of nonzero solutions, and the geometrical structure of the set of nonzero solutions of the system as the number of unknowns tends to infinity.

Properties of easily computable permutations specified in terms of a finite ring are investigated. It is established that problems of parametric identification for this class of permutations are reduced to the resolution of systems of Diophantine equations of sufficiently high degree and to the solution of systems of problems of taking discrete logarithms.

Basic finite-automaton characteristics are established for the class of all linear automata and information-lossless automata over a ring. The complexities of solving problems of parametric identification and initial-state identification are analyzed. The sets of fixed points for mappings realized by initial automata are characterized. Canonical forms are proposed for linear automata over the ring.

The ranks of cyclic and negacyclic codes over the finite chain ring R as well as their minimal generating sets are defined, and then the expression forms we presented by studying the structures of cyclic and negacyclic codes over the finite chain ring R. Through the paper, it is assumed that the length of codes n can not be divided by the characteristic of R.

Given a finite set of polynomial-exponential, multivariate, and vector-valued sequences, we show how the smallest linear shift-invariant set containing the data trajectories can be written as the solution set of a system of linear difference equations with constant coefficients. The resulting representation is known as the most powerful unfalsified model (MPUM) in behavioral systems theory. We address the case where the components of the given sequences take their values in a field, as well as the case where these values belong to a finite ring of the form $${{\mathbb{Z}}/m{\mathbb{Z}}}$$ for an integer m > 1.

In this study, we consider the Kondo temperature and differential conductance for an Aharonov-Bohm ring with an embedded quantum dot connected with noncollinear ferromagnetic leads. Starting from the tight-binding model, we propose an equivalent Anderson model, in which the density of states depends on the Aharonov-Bohm phase. By applying the poor man’s scaling approach to the Hamiltonian, we derive the dependences of the Kondo temperature and differential conductance on the Aharonov-Bohm phase, spin polarization, angle of magnetic moment, and asymmetry parameters. We show conditions for the nonmonotonic behavior of the differential conductance in terms of the spin splitting and Aharonov-Bohm phase. In addition, by extending the model to the case of a finite ring size, we show that the Kondo temperature crucially depends on ring size, but the properties of the scaled temperature are similar to ones for the small ring-size limit.

The properties of the class of information-lossless automata represented by recurrent relations over a finite ring are investigated. For these automata, the structure of classes of equivalent states is investigated, problems of parametric identification and identification of the initial state are solved, and the variation in the behavior of such automata as a result of variation of their parameters or initial states is characterized.

It is well-known that for a lattice-finite order Λ over a complete discrete valuation domain, the radical of Λ-lat (the category of Λ-lattices) is nilpotent modulo projectives. Iyama has shown that this property merely depends on the combinatorial data given by the Auslander–Reiten quiver of Λ. Moreover, he established a criterion for a finite (symmetrizable) translation quiver Q to be the Auslander–Reiten quivers of an order Λ. We improve his characterization by showing that the remaining conditions on Q can be replaced by the existence of an additive function on the vertices of Q (Theorem 4). Our proof rests on a functorial theory of ladders, expressing the Auslander–Reiten structure of Λ-lat by means of an adjoint pair of functors L⊣L− in the homotopy category of two-termed complexes over Λ-lat.

Up to derived equivalence, the representation-finite self-injective algebras of class A n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Mobius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Mobius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Mobius algebras.

It is shown that a finite ring R is a Frobenius ring if and only if $_R(R/\hbox {Rad}\, R)\cong \hbox {Soc}\, (_RR)$ . Other combinatorial characterizations of finite Frobenius rings are presented which have applications in the theory of linear codes over finite rings.

Let $Aut_kK$ be the automorphism scheme of a finite purely inseparable field extension $k\to K$ (in fact, we consider a wide class of finite ring extensions). Let $A_M$ be the splitting algebra of K (see 2.8) and $K_M={A_M}_{\text red} $ the splitting field of K. It is proved that $Aut_kK$ is integral if and only if $A_M=K_M$ . This may be formulated as a condition on the degree of $K_M$ and generalizes a result of Chase. Surprisingly, $A_M$ may be defined intrinsically, since $\mathrm{Spec} A_M$ is the scheme parametrizing the maximal smooth subgroups of $Aut_kK$ . It is also proved that the desingularization of $Aut_kK$ is the universal maximal smooth subgroup of $Aut_kK$ and coincides with the blowing up along a closed subscheme canonically defined from the action of $Aut_kK$ on $A_M$ .

In this paper we show that a finite nilpotent ring that is not a zero-ring cannot admit a natural duality. In fact, every finite ring having a nilpotent subring (which is nilpotent of class ≥ 2) is not dualizable.

A detailed study of digital (t, m, s)-nets and digital (T,s)-sequences constructed over finite rings is carried out. We present general existence theorems for digital nets and sequences and also explicit constructions. Special attention is devoted to the case where the finite ring is a residue class ring of the integers. This study is motivated by the problem of numerical integration of multivariate Walsh series by quasi-Monte Carlo methods, for which we also provide a general error bound.

The Chinese remainder theorem is a fundamental technique widely employed in digital signal processing for designing fast algorithms for computing convolutions. Classically, it has two versions. One is over a ring of integers and the second is over a ring of polynomials with coefficients defined over a field. In our previous papers, we developed an extension to this well-known theorem for the case of a ring of polynomials with coefficients defined over a finite ring of integers. The objective was to generalize number-theoretictransforms, which turn out to be a special case of this extension. This paper focuses on the extension of the Chinese remainder theorem for processing complex-valued integer sequences. Once again, the present work generalizes the complex-number-theoretic transforms. The impetus for this work is provided by the occurrence of complex integer sequences in digital signal processing and the desire to process them using exact arithmetic.

In [3] and also in [4], Freese and McKenzie prove that a finitely generated congruence modular variety is residually small if and only if it is residually less than n for some n < ~o. In fact, they showed that ifA generates a residually small congruence modular variety and [A] = m, then "//(A) is residually less than the bound l! + m where l = m .... 3. They obtained this result by associating to ~'(A) a finite ring with unit, R = R(~(A), m), and relating the size of the subdirectly irreducible algebras in "f/"(A) to the size of subdirectly irreducible R-modules. Their final bound is based on two estimates. First they estimate that tR[ < / = m m'~ ~. Then they show that ifR is a ring of cardinality l, the subdirectly irreducible R-modules are bounded by l!. Using these estimates and applying modular commutator theory leads them to their result. In [1], we asked whether there is a function f such that for any residually small variety f~ with the CEP that is generated by an algebra of cardinality m, one has that ~ is residually less than f(m). This question was answered affimatively by E. Kiss in [2]. His proof does not produce a suitable function and no choice for f is known. We will show that, if R is finite, no subdirectly irreducible R-module can have larger cardinality than IR]. This has the effect of removing the factorial sign in Freese and McKenzie's result. It has a second consequence of providing a tight bound on the size of subdirectly irreducibles in finitely generated, congruence modular varieties with the CEP.

Testing large VLSI circuits is a difficult and challenging problem for designers. Large unstructured circuits are often impossible to test. The number of test vectors also tend to be large and difficult to generate using automated tools for testing. In this paper, we have investigated the testing of systolic arrays built from a finite ring cell that has been proposed recently for digital signal processing functions. The cell has been shown to allow encoding, decoding and general inner product type computations for residue number system applications, with considerable advantages over equivalent binary implementation. As a further feature, we show, in this paper, that an array of such cells is remarkably easy to test for stuck-at faults. Generating test vectors for these arrays is also straightforward.

In [lJ Baldwin and Berman showed that for varieties ~ with DPC (definable principal congruences) certain results of Taylor concerning residually small varieties could be sharpened. Their question as to whether every variety generated by a finite algebra has DPC was answered in the negative in [2]; however the question remained open for varieties with permutable congruences. The study of DPC became even more interesting when McKenzie [4] proved that this property could be used, in certain cases (such as a variety generated by a para-primal algebra), to give an easy proof of the finite axiomatizablity of the variety. McKenzie then showed that among lattices only the distributive varieties have DPC, and states that the question of whether varieties generated by a finite group or ring have DPC is open. In the first section we point out that a variety ~ has DPC iff the free algebra on countably many generators in ~ has SDPC (strongly definable principal congruences), hence a variety generated by a class Y{ of algebras has DPC iff the quasi-variety generated by 5g has DPC. In the second section a finite ring R is constructed such that the variety generated by R does not have DPC. In the third section we prove that if the variety generated by a finite group G has DPC then G must be nilpotent; on the other hand if G is nilpotent class 2 and finite then indeed it generates a variety with DPC. It follows that the properties of having DPC and being finitely axiomatizable are independent for quasi-varieties generated by a finite group. Finally Baldwin's theorem (3) that the variety of all groups of exponent 3 has DPC is shown to be best possible for Burnside varieties.

In this note we study rings having only a finite number of non isomorphic uniform modules with non zero socle. It is proved that a commutative ring with this property is a direct sum of a finite ring and a ring of finite representation type. In the non commutative case we show that most P.I. rings having only a finite number of non isomorphic modules with non zero socle are in fact artinian.

This paper presents design details for a bit-level systolic cell that has been recently introduced for imple menting digital signal processing (DSP) operations over finite rings. This paper concentrates on the efficient construction of the basic cell, using a 3µ p-well CMOS technology. The design uses a 5-bit, 32-word dynamic ROM as the main computational element, and details of all elements of the cell are discussed with regard to minimizing the (Area. Period) product. The final cell design and simulation details are compared with a pipelined gated full-adder, designed in the same technology, which represents the most common type of binary bit-level systolic cell. It is shown that the (Area. Period) product of the binary cell is 68% greater than that of the finite ring cell, but the power of the finite ring cell, in implementing fixed coefficient inner product multiplications, is much greater than that of the binary cell. There are also advantages associated with the reduction of the connectivity across the dynamic range, including clock-skew reduction, ease of testing, and fault detection. The conclusion is that in certain classes of DSP operations, this new cell can offer more than an order of magnitude improvement in (Area.Period) product of complete bit-level systolic arrays, over its binary counterpart.

Near rings without zero divisors, and a dual structure, near codomains, are studied. It is shown that a near ring is a near field if and only if it is an integral near ring, a near codomain, and has a non-zero distributive element. If the additive group (N, +) of a near integral domainN is cohopfian, then (N, +) possesses a fixed point free automorphism which is either torsion free or of prime order. This generalizes a well-known theorem of Ligh for finite near integral domains. A result ofGanesan [1] on the non-zero divisors in a finite ring is generalized to near rings.