An algorithm for commutative semigroup algebras which are principal ideal rings
نویسندگان
چکیده
Associative and commutative algebras with identity have various well-known applications. In particular, many classical codes are ideals in commutative algebras (see [4], [12] for references). Computer storage, encoding and decoding algorithms simplify if all these codes have single generator polynomials. Thus it is of interest to determine when all ideals of an algebra are principal. In [5] Decruyenaere, Jespers and Wauters characterized commutative semigroup algebras with identity which are principal ideal rings. In the more general case of noncommutative algebras all principal ideal semigroup algebras have been described by Jespers and Okninski [9]. For preliminaries on semigroup rings and other related constructions we refer to [13] and [10]. In this paper we develop an algorithm which, given a presentation for a commutative semigroup S and the characteristic of a field k, decides whether the semigroup algebra k[S] is a principal ideal ring with identity. This builds
منابع مشابه
The principal ideal subgraph of the annihilating-ideal graph of commutative rings
Let $R$ be a commutative ring with identity and $mathbb{A}(R)$ be the set of ideals of $R$ with non-zero annihilators. In this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of $R$, denoted by $mathbb{AG}_P(R)$. It is a (undirected) graph with vertices $mathbb{A}_P(R)=mathbb{A}(R)cap mathbb{P}(R)setminus {(0)}$, where $mathbb{P}(R)$ is...
متن کاملGroebner Bases in Non-Commutative Algebras
INTRODUCTION Recently, the use of Groebner bases and Buchberger algorithm [BUC1,2,4] has been generalised from the case of commutative polynomials to finitely generated algebras R over a field k, R = k, s.t. for each i < j, for some cij ∈ k, for some commutative polynomial pij ∈ k[X1,...,Xn], one has xj xi cij xi xj = pij(x1,...,xn). The first results in this direction were due to Ga...
متن کاملCodes through Monoid Rings and Encoding
Cazaran and Kelarev [2] have given necessary and sufficient conditions for an ideal to be the principal; further they described all finite factor rings Zm[X1, · · · , Xn]/I, where I is an ideal generated by an univariate polynomial, which are commutative principal ideal rings. But in [3], Cazaran and Kelarev characterize the certain finite commutative rings as a principal ideal rings. Though, t...
متن کاملOn Commutative Reduced Baer Rings
It is shown that a commutative reduced ring R is a Baer ring if and only if it is a CS-ring; if and only if every dense subset of Spec (R) containing Max (R) is an extremally disconnected space; if and only if every non-zero ideal of R is essential in a principal ideal generated by an idempotent.
متن کاملON COMMUTATIVE GELFAND RINGS
A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007