Localizations of Dedekind prime rings
Localizations of Dedekind prime rings
复制标题
戴德金素环的本地化
DOI：
10.1016/0021-8693(72)90002-6
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发表时间：
1972
影响因子：
0.9
通讯作者：
J. Kuzmanovich
中科院分区：
文献类型：
--
作者：
J. Kuzmanovich
本文的目的是得到具有典型商环Q（Q是单Artin环）的Dedekind素环R的局部化和整体化定理。所得到的R的局部化由R的每个极大双边理想M的一个子环RM和一个附加的简单覆盖环S组成，当R有界时，该覆盖环S与Q重合。全局化结果表明：（<$R m）<$S= R，R-模的同态<$M是one-one（onto）当且仅当所有的“局部化”同态<$M，和<$S都是oneone（onto）。环T= R R M是有界Dedekind素环，其双边理想格同构于R的双边理想格（保剩余环）.此外，每一个整右R-理想唯一地可表示为交J <$K，其中J（K）是整右S-理想（T-理想）。每个环R M都是局部的，因为它是一个有界主理想环，具有唯一的极大双侧理想M（Jacobson根），并且每个双侧理想都是M的幂。R M在M的完备化自然地同构于R在M的完备化。M的消去集C（M）被定义为{B <$R：bx <$M蕴涵x <$M}。R M的构造包括证明C（M）是R的正则元素的乘法闭集，关于它R满足Ore条件，然后定义R M为{ab− 1：a C（M）}。环S被定义为{q R对于R的某个非零双边理想B}。S是单Dedekind素环，且整右S-理想格同构于完全忠实右R-理想格。在本文中，所有的环理论条件，如阿廷，诺特，和遗传将是双边的，除非另有说明。
The purpose of this paper is to obtain localization and globalization theorems for a Dedekind prime ring R with classical quotient ring Q (Q is simple Artinian). The localizations of R which are obtained consist of one subring R M for each maximal two-sided ideal M of R, and one additional simple overring S which coincides with Q when R is bounded. The globalization results state that (∩ R m)∩ S= R, and that a homomorphism ƒ of R-modules is one-one (onto) if and only if all of the “localized” homomorphisms ƒ M, and ƒ S are oneone (onto). The ring T=∩ R M is a bounded Dedekind prime ring whose two-sided ideal lattice is isomorphic (preserving residue rings) to the two-sided ideal lattice of R. Furthermore, every integral right R-ideal is uniquely expressible as an intersection J∩ K where J (K) is an integral right S-ideal (T-ideal). Each of the rings R M is “local” in the sense that it is a bounded principal ideal ring with a unique maximal two-sided ideal M (the Jacobson radical), and every two-sided ideal is a power of M. The completion of R M at M is in a natural way isomorphic to the completion of R at M. The cancellation set of M, C (M), is defined to be {b ϵ R: bx ϵ M implies x ϵ M}. The construction of R M consists of showing that C (M) is a multiplicatively closed set of regular elements of R with respect to which R satisfies the Ore conditions and then defining R M to be {ab− 1: a ϵ R, b ϵ C (M)}. The ring S is defined to be {q ϵ Q: qB⊂ R for some nonzero two-sided ideal B of R}. S is a simple Dedekind prime ring and the lattice of integral right S-ideals is isomorphic to the lattice of completely faithful right R-ideals. Throughout this paper all ring theoretic conditions such as Artinian, Noetherian, and hereditary will be two-sided unless otherwise specified.