论文标题
减少通勤环的最小条件受限
Restricted minimum condition in reduced commutative rings
论文作者
论文摘要
我们说，如果对于r，r/i中的所有基本理想I是Artinian环，则可以满足限制的最小值（RM）条件。我们将重点放在Noetherian减少环上，因为在此设置中，RM域的已知结果良好。但是，正如我们将要显示的那样，RM环不必是noeetherian，并且可能具有nilpotent元素。 RM环理论的经典结果之一是，对于Noetherian域，RM条件最多与具有Krull维度相对应。我们将证明这可以推广到缩小Noetherian环，因此证明与曲线相对应的仿射环为RM。我们将举一个例子，表明降低环的假设不是多余的。 我们将证明CDR域是RM，这将使我们能够对Dedekind域进行新的特征。将提供各种环的RM环的示例。特别是，我们将显示一个多项式R [X]的环是RM，并且仅当R减少Artinian环时。我们将研究RM环与UFD之间的关系。
We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM domains generalize well. However, as we will show, RM rings need not be Noetherian and may have nilpotent elements. One of the classic results in the theory of RM rings is that for Noetherian domains RM condition corresponds to having Krull dimension at most one. We will show that this can be generalized to reduced Noetherian rings, thus proving that affine rings corresponding to curves are RM. We will give examples showing that the assumption that the ring is reduced is not superfluous. We will prove that CDR domains are RM and this will allow us to give a new characterization of Dedekind domains. Examples of RM rings for various classes of rings will be given. In particular, we will show that a ring of polynomials R[x] is RM if and only if R is reduced Artinian ring. And we will study the relation between RM rings and UFDs.
