论文标题
驯服和狂野的素数直接戒指的直接产品
Tame and wild primes in direct products of commutative rings
论文作者
论文摘要
几十年来，对通勤环的无限直接产物的所有主要理想的结构（例如，几十年来）一直是一个挑战性的问题。在本文中，在这方面取得了新的进步。我们观察到，在非零环的无限直接产品中，我们称之为驯服的素数和野生素数有两种不同类型的理想。在主要结果中，我们证明了驯服素数的集合是素数的开放式亚气管，并且仅当索引集是无限的时，该方案是非属于的。作为一种应用，当它只有包含直接总和理想的情况下，质量理想才是野外的素数。接下来，我们表明，索引集的（非主要）超滤器（至少$ 2^{\ Mathfrak {c}} $ wild Primes，$ \ Mathfrak {C c} $是Continuum的持卡率）。该描述具有重要的结果：如果直接产品环具有野生质数，那么一组野生素数是无限的（无数）。还研究了无限直接乘积环的原始光谱的连接组件。我们观察到，与主要理想一样，在这个空间中，有两种类型的连接组件，驯服的组件和野性组件。
A complete understanding of the structure of all prime ideals of an infinite direct product of commutative rings (e.g. in terms of more specific objects) has remained a challenging problem for decades. In this article, new advances have been made in this regard. We observe that in an infinite direct product of nonzero rings there are two different types of prime ideals, that we call tame primes and wild primes. Among the main results, we prove that the set of tame primes is an open subscheme of the prime spectrum, and this scheme is non-affine if and only if the index set is infinite. As an application, a prime ideal is a wild prime if and only if it contains the direct sum ideal. Next, we show that an uncountable number of (wild) primes of an infinite direct product ring are induced by the (non-principal) ultrafilters of the index set (at least $2^{\mathfrak{c}}$ wild primes, $\mathfrak{c}$ is the cardinality of the continuum). This description has an important consequence: if a direct product ring has a wild prime, then the set of wild primes is infinite (uncountable). The connected components of the prime spectrum of an infinite direct product ring are also investigated. We observe that, like for prime ideals, there are two types of connected components in this space, tame ones and wild ones.
