Convex Polytopes and Factorization Properties in Generalized Power Series Domains
Convex Polytopes and Factorization Properties in Generalized Power Series Domains
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广义幂级数域中的凸多面体和因式分解性质
DOI：
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发表时间：
2008
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通讯作者：
David E. Rush
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作者：
G. Brookfield;David E. Rush
它示出了如何关联到任何多面体，是不是一个单一的和任何领域K，一个交换的整体域D，没有不可约的元素，这是不是前Schreier。整环D是K上的广义幂级数环。设R是具有商域K的整环.回想一下，如果a不是R的两个非单位元的乘积，则称a ∈ R \ {0}是不可约的或原子，如果对于所有B，c ∈ R，a是素数，则称a是素数。|BC表示A| B或A| C.要证明任何素元都是不可约的是很容易的，而且人们已经做了很多研究来探讨什么时候匡威命题成立。例如，在任何pre-Schreier整环中，所有的不可约元素都是素的，因此我们回想一下定义：整环R中的元素a是素的，如果当a整除bc与R中的B和c时，则a = B′c′，对于某个B′，c′ ∈ R，其中B′整除B，c′整除c。一个整环，其中每个元素都是原始的，被称为pre-Schreier。（一个Schreier域是一个准Schreier域，它也是整闭的。许多学者都对这类环进行了研究，例如[6]、[7]、[9]、[13]、[18]、[22]。在一个pre-Schreier整环中，每个不可约元都是素元，这是直接的。另一方面，也有一些整环的例子不是pre-Schreier整环，但其中每个不可约元都是素的（见[18，例3.7]）。(See[1]这是一个比较，这些属性和几个相关的。在[21]中，W. C. Waterhouse证明，如果每个二次多项式f ∈ R[X]在K[X]中因式化为线性多项式，则R中的每个不可约元都是素元。这个结果和pre-Schreier条件之间的关系在[18]中进行了探讨。
It is shown how to associate to any polytope that is not a simplex and any field K, a commutative integral domain D which has no irreducible elements and which is not pre-Schreier. The integral domain D is a generalized power series ring over K. Let R be an integral domain with quotient field K. Recall that a ∈ R \ {0} is said to be irreducible, or an atom, if a is not the product of two nonunits of R, and that a is said to be prime if, for all b, c ∈ R, a|bc implies a|b or a|c. It is easy to show that any prime element is irreducible, and much research has been done into the question of when the converse is true. For example, in any pre-Schreier domain, all irreducible elements are prime, and so we recall the definition: An element a of an integral domain R is primal if, whenever a divides bc with b and c in R, then a = b′c′ for some b′, c′ ∈ R where b′ divides b and c′ divides c. An integral domain in which each element is primal is said to be pre-Schreier. (A Schreier domain is a pre-Schreier domain which is also integrally closed.) Such rings have been studied by many authors, for example, [6], [7], [9], [13], [18], [22]. It is immediate that, in a pre-Schreier domain, each irreducible element is prime. On the other hand, there exist examples of integral domains which are not pre-Schreier, but in which each irreducible element is prime (see [18, Example 3.7]). (See also [1] for a comparison of these properties and several related ones.) In [21], W. C. Waterhouse shows that, if each quadratic polynomial f ∈ R[X] factors into linear polynomials in R[X] whenever it factors into linear polynomials in K[X], then every irreducible element in R is prime. The relation between this result and the pre-Schreier condition was explored in [18]