Serial rings and finitely presented modules
Serial rings and finitely presented modules
复制标题
串行环和有限呈现模块
DOI：
10.1016/0021-8693(75)90074-5
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发表时间：
1975
影响因子：
0.9
通讯作者：
R. Warfield
中科院分区：
文献类型：
--
作者：
R. Warfield
一个模是串行的，如果它的子模关于包含是线性有序的。环R称为左串行，如果RR是串行模的直和，R称为串行，如果它是左和右串行。序列环提供了交换赋值环和广义唯一代数的自然推广。本文证明了串行环上的一个可表示模是串行模的直和。应用给出了半完美半遗传环和交换Noether环上的“局部序列”代数的结构。4.给出了Noether序列环的相当完备的结构理论。第一节我们回顾了半完全环的基本性质。如果R是半完全的，M是一个FP R-模，S是一个简单的R-模，我们引入两个数值不变量Gen（M; S）和Rel（M; S），我们可以用它们来精炼通常用生成元和关系来表述的语句。我们用它来给出半完全环的结构理论，其中每个生成的左理想都是主环（1.14）。我们还证明了（1.4和1.5），半完全环上的每个稳定同构类的n-生成模包含一个唯一的（直到同构）极小元。（回想一下，如果存在射影P和Q使得A@ P E B@ Q，则A和B是稳定同构的。）这些结果在第2节中被用来给出Auslander和布里杰[I]关于FP右模和FP左模之间的对偶理论的一个精化形式。在第三节中，我们证明了序列环上的每一个双表示模都是序列模的直和。我们实际上展示了更多。假设一个模是局部的，如果它有一个唯一的极大真子模，并且M是局部表现的（LP），如果存在一个正合序列P-+ Q+：II+ 0，其中P和Q是局部投射的。然后，我们证明（2.6和3.4），
A module is serial if its submodules are linearly ordered with respect to inclusion. A ring R is called left serial if RR is a direct sum of serial modules, and R is called serial if it is both left and right serial. Serial rings provide a natural generalization of both commutative valuation rings and generalized uniserial algebras. In this paper, it is shown that a finitely presented module over a serial ring is a direct sum of serial modules. Applications are given to the structure of semiperfect semihereditary rings and “locally serial” algebras over commutative Noetherian rings.. 4 fairly complete structure theory is given for Noetherian serial rings.In the first section we review the basic properties of semiperfect rings. If R is semiperfect, M is a finitely presented (FP) R-module, and S is a simple R-module, we introduce two numerical invariants, Gen (M; S) and Rel (M; S), which we can use to refine statements usually made in terms of generators and relations. We use this to give the structure theory of semiperfect rings for which every finitely generated left ideal is principal (1.14). We also show (1.4 and 1.5) that every stable isomorphism class of finitely generated modules over a semiperfect ring contains a unique (up to isomorphism) minimal element.(Recall that A and B are stably isomorphic if there are projective P and Q such that A@ P E B@ Q.) These results are used in Section 2 to give a refined form of the duality theory of Auslander and Bridger [I], between FP right modules and FP left modules. In Section 3 we prove that every finitely presented module over a serial ring is a direct sum of serial modules. We actually show more. Say that a module is local if it has a unique maximal proper submodule, and M is locally presented (LP) if there is an exact sequence P-+ Q+: II+ 0 in which P and Q are local projectives. We show, then (2.6 and 3.4), that the following