A note over constructions of cyclic codes over certain semigroups

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.

The mini workshop "Arithmetic of group rings" was attended by 16 participants from Belgium, Brazil, Canada, Germany, Hungary, Israel, Italy, Romania and Spain. The expertise was a good mixture between senior and young researchers. It was a very stimulating experience and thesize of the group allowed excellent discussions amongst all participants. Very fruitful were the problem sessions, resulting in the problems listed at the end of this report. The main highlights of the conference were: The complete calculation of the projective Schur subgroup of the Brauer group by Aljadeff and del Rio. Hertweck's solution of the first Zassenhaus conjecture for finite metacyclic groups. the description of special subgroups of the unit group of integral group rings, such as the hypercentre and the finite conjugacy centre, and the relation with respect to the normalizer of the trivial units. discussion of the present state of art via several survey talks presented and the problem sessions The group G determines its integral group ring \mathbb Z G and its group V(\mathbb Z G) of normalized units. Several talks addressed the interplay of the cohomological properties of these three objects. Further topics included twisted group rings, group rings over local rings, polynomial growth and identities, orders and semigroup rings, Lie structure, representation-theoretic and algorithmic methods.

Let S be a cancellative monoid with quotient group of torsion-free rank a.We show that the monoid ring R[S] is a Hilbert ring if and only if the polynomial ring R[{ X, },s/] is a Hilbert ring, where |/| = a.Assume that R is a commutative unitary ring and G is an abelian group.The first research problem listed in [K, Chapter 7] is that of determining equivalent conditions in order that the group ring R[G] should be a Hilbert ring.Matsuda has considered this question in [Ml, 6].His results show that for G finitely generated, R[G] and R are simultaneously Hilbert rings; if G is not finitely generated, then R[G] a Hilbert ring implies that R is a Hilbert ring, but the converse fails.In [M2, 5] Matsuda considers briefly the corresponding Hilbert-ring-characterization problem for a monoid ring R[S], where S is torsion-free and cancellative.Let a be the torsion-free rank of G and let {Xj)lsl be a set of indeterminates over R of cardinality a.In Corollary 1 we show that R[G] and P[{^,},e/] are simultaneously Hilbert rings.While the problem of determining equivalent conditions for P[{Ar,}/e/] to be a Hilbert ring has not been completely resolved, it has been worked on extensively [Kr, Go, L, Gl, H], and a significant body of positive results exists concerning this problem.The proof of Theorem 1 suggests the following conjecture: if S is a cancellative commutative monoid with quotient group G, then R[S] is a Hilbert ring if and only if R[G] is a Hilbert ring; this result is established in Theorem 2.All monoids considered are assumed to be commutative, and rings are assumed to be commutative and unitary.The statement of Theorem 1 uses the following terminology.For a cardinal number a, an extension ring T of R is said to be a-generated over R if T is generated over R by a set of cardinality at most a.Theorem 1. Assume that X = {XA is a set of indeterminates of cardinality a over the ring R. Denote by Za the direct sum of a copies of the additive group Z of integers.The following conditions are equivalent.(1) R[X] is not a Hilbert ring.(2) There exists a prime ideal P of R such that R/P admits an a-generated extension ring that is a G-domain, but not afield.(3) The group ring R[Za] is not a Hilbert ring._

Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

Let A⊆B be cancellative abelian semigroups, and let R be an integral domain. We show that the semigroup ring R[B] can be decomposed, as an R[A]-module, into a direct sum of R[A]-submodules of the quotient ring of R[A]. In the case of a finite extension of positive affine semigroup rings, we obtain an algorithm computing the decomposition. When R[A] is a polynomial ring over a field, we explain how to compute many ring-theoretic properties of R[B] in terms of this decomposition. In particular, we obtain a fast algorithm to compute the Castelnuovo–Mumford regularity of homogeneous semigroup rings. As an application we confirm the Eisenbud–Goto conjecture in a range of new cases. Our algorithms are implemented in the Macaulay2 package MonomialAlgebras.

Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications. As they are also of paramount importance in Algebraic Geometry, polynomial rings have been intensively studied. On the other hand, rings of formal power series have been extensively used in “algebroid geometry” and have many properties which are parallel to those of polynomial rings. In the first section of this chapter we shall define formal power series rings and we shall show that the main properties of polynomial rings which have been derived in previous chapters (see, in particular, Vol. I, Ch. I, §§ 16–18) hold also for formal power series rings. In the later sections of this chapter we shall give deeper properties of polynomial rings and, whenever possible, the parallel properties of power series rings.

Let R be a commutative ring with identity and $${\mathbb {N}}_0$$ be the additive monoid of nonnegative integers. We say that a function $$t : {\mathbb {N}}_0 \times {\mathbb {N}}_0 \rightarrow R$$ is a twist function on R if t satisfies the following three properties for all $$n, m, q \in {\mathbb {N}}_0$$ : (i) $$t(0,q) = 1$$ , (ii) $$t(n,m) = t(m,n)$$ , and (iii) $$t(n,m) \cdot t(n + m, q) = t (n, m + q) \cdot t(m, q)$$ . Let $$R[\![X]\!]$$ (resp., R[X]) be the set of power series (resp., polynomials) with coefficients in R. For $$f = \sum _{n=0}^{\infty } a_nX^n$$ and $$g = \sum _{n=0}^{\infty } b_nX^n \in R[\![X]\!]$$ , let $$f+g = \sum _{n=0}^{\infty } (a_n+b_n)X^n$$ , $$f*_tg = \sum _{n=0}^{\infty }(\sum _{i+j = n}t(i,j)a_ib_j)X^n$$ . Then, $$R^t[\![X]\!]:= (R[\![X]\!], +, *_t)$$ and $$R^t[X] := (R[X], +, *_t)$$ are commutative rings with identity that contain R as a subring. In this paper, we study ring-theoretic properties of $$R^t[\![X]\!]$$ and $$R^t[X]$$ with focus on divisibility properties including UFDs and GCD-domains. We also show how these two rings are related to the usual power series and polynomial rings.

We define semigroup semirings by analogy with group rings and semigroup rings. We study the arithmetic properties and determine sufficient conditions under which a semigroup semiring is atomic, has finite factorization, or has bounded factorization. We also present a semigroup-semiring analog (although not a generalization) of the Gauss lemma on primitive polynomials.

Noetherian Semigroup Rings with Several Objects

Two questions on semi-clean group rings

In this paper, we introduce the concept of S‐‐ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of ‐ideals apply to S‐‐ideals and examine their characteristics in various ring constructions, such as homomorphic image rings, quotient rings, cartesian product rings, polynomial rings, power series rings, idealization rings, and amalgamation rings. In noncommutative rings, where S is an m‐system, we define right S‐‐ideals. We demonstrate the equivalence of S‐‐ideals and right S‐‐ideals in commutative rings with identity and provide examples to illustrate the connections between right S‐prime ideals and ‐ideals.

It is well known that a polynomial [Formula: see text] over a commutative ring [Formula: see text] with identity is a zero-divisor in [Formula: see text] if and only if [Formula: see text] has a non-zero annihilator in the base ring, where [Formula: see text] is the polynomial ring with indeterminate [Formula: see text] over [Formula: see text]. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring [Formula: see text] over an associative non-commutative ring [Formula: see text]. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly [Formula: see text] rings and rings with right Property [Formula: see text]. We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].

Let S be a cancellative semigroup. This paper is motivated by the problem of finding a description of semigroup rings K[S] over a field K that are semiprime or prime. Results of this type are well-known in the case of a group ring K[G], cf. [8]. The description, as well as the proofs, involve the FC-centre of G defined as the subset of all elements with finitely many conjugates in G. In [4], [5] Krempa extended the FC-centre techniques to the case of an arbitrary cancellative semigroup S. He defined a subsemigroup Δ(S) of S which coincides with the FC-centre in the case of groups, and can be used to describe the centre and to study special elements of K[S]. His results were strengthened by the author in [7], where Δ(S) was also applied in the context of prime and semiprime algebras K[S]. However, Δ(S) itself is not sufficient to characterize semigroup rings of this type. We note that in [2], [3] Dauns developed a similar idea for a study of the centre of semigroup rings and certain of their generalizations.

<abstract><p>An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $ p, n, r, k, k', m $ up to isomorphism by finite commutative chain rings ($ k' = 1 $). Thus the classification of chain rings is reduced to that of finite commutative chain rings.</p></abstract>

Let $KG$ be the group ring of a group $G$ over a commutative ring $K$ with unity. The rings $KG$ are described for which $xx^\sigma =x^\sigma x$ for all $x=\sum _{g\in G}\alpha _gg\in KG$, where $x\mapsto x^\sigma =~\sum _{g\in G}\alpha _gf(g)\sigma (g)$ is an involution of $KG$; here $f: G\to U(K)$ is a homomorphism and $\sigma$ is an antiautomorphism of order two of $G$.