Serial rings with right Krull dimension one, II
Serial rings with right Krull dimension one, II
复制标题
右克鲁尔尺寸一、二系列环
DOI：
10.1016/0021-8693(88)90243-8
复制
发表时间：
1988
影响因子：
0.9
通讯作者：
M. H. Wright
中科院分区：
文献类型：
--
作者：
M. H. Wright
如果一个模块的子模块在包含范围内线性排序，则该模块是单序列的。如果一个环是单序列右理想的直和，那么它就是右序列。不假设任何链条件。正确的序列环必然是半完美的。具有右克鲁尔维一的非奇异（左和右）串行环的结构在[12]中描述。在本文中，我们描述了右克鲁尔维一的任意串行环的结构。我们还给出了一个例子来表明，克鲁尔维一的串行环上的统一模块不一定是单串行的。因此，在诺特序列环和克鲁尔一维非奇异序列环的研究中非常有用的事实 [12, 131] 不再可用。 We shall show that the local projective modules over a serial ring of Krull dimension one are naturally organized into “cliques”(see definitions below).这些产生了基本的串行环：只有一个派系的串行环。基本序列环可分为四类之一，具体取决于它所产生的派系的特征。这是第 1 节的主题。在第 2 节中，我们根据由其团簇产生的基本环来描述克鲁尔维一的任意串行环的结构。本地模块是具有唯一最大真子模块的模块。本地元素是生成本地模块的元素。有关 Krull 维度和关键模块的详细信息，请参阅[6]。如果环是 Artinian 且两侧都是单射热电联环，则该环是 QF。如果 R 是右 R 模内射且具有 fg 本质右脚的半完美，则 R 是右 PF。如果存在最小忠实模块（从某种意义上说，它是每个忠实模块的直接被加数），则 R 是正确的 QF-3。 Z (M) 表示模块 M 的奇异子模块。给定简单右 R 模块 U 和 V，如果存在满足 M/MJz V 和 MJk EU 的单序列模块 M，则 U 被称为 V 的（第 k 度）后继者（并且 V 是 U 的第 k 度前身）[131。 V 的（一级）后继（如果存在）用 U= a (V) 表示。 99
A module is uniserial if its submodules are linearly ordered under inclusion. A ring is right serial if it is a direct sum of uniserial right ideals. No chain conditions are assumed. A right serial ring is necessarily semiperfeet. The structure of a nonsingular (left and right) serial ring with right Krull dimension one was described in [12]. In this paper, we describe the structure of an arbitrary serial ring of right Krull dimension one. We also give an example to show that a uniform module over a serial ring of Krull dimension one need not be uniserial. Thus a fact which was very useful in the study of Noetherian serial rings and nonsingular serial rings of Krull dimension one [12, 131 is no longer available. We shall show that the local projective modules over a serial ring of Krull dimension one are naturally organized into “cliques”(see definitions below). These give rise to elementary serial rings: serial rings which have only one clique. An elementary serial ring falls into one of four categories, depending on characteristics of the clique from which it arises. This is the subject matter of Section 1. In Section 2 we describe the structure of an arbitrary serial ring of Krull dimension one in terms of the elementary rings arising from its cliques. A local module is one with unique maximal proper submodule. A local element is one which generates a local module. For details on Krull dimension and critical modules, see [6]. A ring is QF if it is Artinian and an injective cogenerator on both sides. R is right PF if R is injective as a right R-module and semiperfect with fg essential right socle. R is right QF-3 if there is a minimal faithful module (in the sense that it is a direct summand of every faithful module). Z (M) denotes the singular submodule of the module M. Given simple right R-modules U and V, U is called a (k-th degree) successor of V (and V a k-th degree predecessor of U) if there exists a uniserial module M such that M/MJz V and MJk EU [131. The (first degree) successor of V, when it exists, is denoted by U= a (V). 99
丸林秀俊：代数通讯 17. (1989)
DOI：
--
发表时间：
--
期刊：
影响因子：
--
作者：
通讯作者：
--