Commutative Rings and Finite Fields
نویسنده
چکیده
منابع مشابه
Self-dual codes over commutative Frobenius rings
We prove that self-dual codes exist over all finite commutative Frobenius rings, via their decomposition by the Chinese Remainder Theorem into local rings. We construct non-free self-dual codes under some conditions, using self-dual codes over finite fields, and we also construct free self-dual codes by lifting elements from the base finite field. We generalize the building-up construction for ...
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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...
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Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exist unit elements $u,v$ of $R$ such that $x+uyv$ is a unit of $R$. In this paper, basic properties of $Gamma(R)$ are studied. We investigate connectivity and the girth of $Gamma(R)$, where $...
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Linear codes over finite rings with identity have recently raised a great interest for their new role in algebraic coding theory and for their successful application in combined coding and modulation. Thus, in this paper we address the problems of constructing of new cyclic, BCH, alternant, Goppa and Srivastava codes over local finite commutative rings with identity. These constructions are ver...
متن کاملCodes over affine algebras with a finite commutative chain coefficient ring
We consider codes defined over an affine algebra A = R[X1, . . . , Xr]/ 〈t1(X1), . . . , tr(Xr)〉, where ti(Xi) is a monic univariate polynomial over a finite commutative chain ring R. Namely, we study the A−submodules of A (l ∈ N). These codes generalize both the codes over finite quotients of polynomial rings and the multivariable codes over finite chain rings. Some codes over Frobenius local ...
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Linear network coding over finite fields is a wellstudied problem. We consider the more general setting of linear coding for directed acyclic networks with finite commutative ring alphabets. Our results imply that for scalar linear network coding over commutative rings, fields can always be used when the alphabet size is flexible, but other rings may be needed when the alphabet size is fixed. W...
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