Gröbner bases for polynomial ideals over commutative regular rings
Gröbner bases for polynomial ideals over commutative regular rings
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交换正则环上多项式理想的 Gröbner 基
DOI：
10.1007/3-540-51517-8_137
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
V. Weispfenning
中科院分区：
文献类型：
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作者：
V. Weispfenning
介绍。1965年Buchberger引入的Gr6bner基计算方法为解决关于场上多项式环理想的许多基本问题提供了算法来源（参见[B2]的综述）。该方法的成功刺激了将该方法扩展到其他地环（以及非交换多项式，参见[M]，[ALl],[KW]）的持续研究。整数环Z为研究更一般类型的地环（如欧几里得环和主理想域）提供了一个自然的起点（参见[BI]和此处引用的文章，[KK]和[Pal]）。关于Gr~ bner基构造的更全面的公理化方法[j]， [j]。X，]在[KNI]和[Moe]中表示；它们基于各自的假设，即R的理想存在GrSbner基构造，并且可以在R [X]中计算协同子模块的基。本文研究的地环是交换正则环，它与以往研究的环有两个本质上的区别：它们一般都不是整域，它们一般都不是noether环。然而，它们与域有一个共同的特征：每个元素都有一个唯一的准逆。特别地，任何域的直积都是交换正则环。另一方面，它们推广了布尔环。因此，它们可以被解释为域和布尔代数的“组合”。这一观点得到了交换正则环作为域的布尔积的成熟表示的支持，或者等价地作为布尔空间上的域束中的全局部分环（参见[Pi]，[W]）。这种表示扩展到可交换正则环R上的多项式环，事实上，扩展到R上代数几何中产生的大多数环和模（参见[SW]）。特别地，任何有限的正倍数
INTRODUCTION. The method of Gr6bner basis computations introduced by Buchberger in 1965 has provided a source of algorithms for solving many basic questions concerning ideals in polynomial rings over fields (see [B2] for a survey). The success of the method has stimulated ongoing research on extensions of the method to other ground rings (and also to non-commutative polynomials, see [M],[ALl,[KW]). The ring Z of integers formed a natural starting point for the study of more general classes of ground rings such as Euclidean rings and principal ideal domains (see [BI] and the articles quoted there,[KK] and [Pal). More comprehensive axiomatic approaches to Gr~ bner basis constructions in R [XI,.. X,] are presented in [KNI] and [Moe]; they are bases on the respective hypotheses, that a GrSbner basis construction is available for the ideals of R, and the bases for modules of syzygies can be computed in R [X].The ground rings studied in this paper, commutative regular rings, differ from most of the rings studied previously in two essential respects: They are in general far from being integral domains, and they are in general not Noetherian. Nevertheless, they share one feature with fields: Every element has a unique quasi-inverse. In particular, any direct product of fields is a commutative regular ring. On the other hand, they generalize Boolean rings. Accordingly, they may be construed as" compositions" of fields and Boolean algebras. This view is supported by the well-established representation of commutative regular rings as Boolean products of fields, or equivalently as rings of global sections in sheaf of fields over a Boolean space (see [Pi],[W]). This representation extends to polynomial rings over a commutative regular ring R, and in fact to most of the rings and modules arising in the algebraic geometry over R (see [SW]). In particular, any finite direct multiple