Global dimension in serial rings
Global dimension in serial rings
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DOI：
10.1016/0021-8693(85)90069-9
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发表时间：
1985-11
影响因子：
0.9
通讯作者：
W. Gustafson
中科院分区：
文献类型：
--
作者：
W. Gustafson
设R是一个不可分解的基本系列artin环，带有根r。众所周知，我们可以对不可分解射影左 R 模的完整集合 P,,..., P, 进行编号，使得 PJrP, ErP,~, lr2Pip, 对于 i= 2,..., n，并且如果 r P,# 0，则 P,/r P, z r P,,/r2P,,。已知 R 的许多属性是由整数 ci= 长度 (P,) 决定的。设 [j] 表示 j 模 n 的最小正余数，只要 j 是正整数。因此，[n]=n。令S,= P;/rP,，由此S,,...,S, 是非同构简单左R 模块的完整集合。那么P, 的组成因子是（从上到下）Si, ScI+ l],-, SCl+ c.,-11。 R 的 Loewy 长度为 LL (R)= max (ci}。它是 r LL (R)= 0 的最小正整数。我们这里的目的是证明
Let R be an indecomposable basic serial artin ring with radical r. As is well known, we can number a full set P,,..., P, of indecomposable projective left R-modules so that PJrP, ErP,~, lr2Pip, for i= 2,..., n, and if r P,# 0, then P,/r P, z r P,,/r2P,,. Many properties of R are known to be determined by the integers ci= length (P,). Let [j] denote the least positive residue of j modulo n, whenever j is a positive integer. Thus,[n]= n. Let S,= P;/rP,, whence S,,..., S, is a full set of nonisomorphic simple left R-modules. Then the composition factors of P, are (from the top down) Si, ScI+ l],-, SCl+ c.,-11. The Loewy length of R is LL (R)= max (ci}. It is the least positive integer with r LL (R)= 0. Our purpose here is to prove