Torsion-Free and Divisible Modules Over Non-Integral-Domains
Torsion-Free and Divisible Modules Over Non-Integral-Domains
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非积分域上的无扭可分模块
DOI：
10.4153/cjm-1963-016-1
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发表时间：
1963
期刊：
影响因子：
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通讯作者：
L. Levy
中科院分区：
文献类型：
--
作者：
L. Levy
在试图将挠的概念推广到比交换整环更一般的环时，我们注意到的第一件事是，如果定义逐字延续，整环是唯一具有无挠模的环。因此，如果m是环上任何右模M的元素，该环包含一对非零元素x和y，使得xy = 0，则mx = 0或（mx）y = 0。第二个困难出现在非交换的情况下：是否扭转元素的集合M形成一个子模？对于任意的非交换整环，这个问题的答案甚至不会是“是”。
In trying to extend the concept of torsion to rings more general than commutative integral domains the first thing that we notice is that if the definition is carried over word for word, integral domains are the only rings with torsion-free modules. Thus, if m is an element of any right module M over a ring containing a pair of non-zero elements x and y such that xy = 0, then either mx = 0 or (mx)y = 0. A second difficulty arises in the non-commutative case: Does the set of torsion elements of M form a submodule? The answer to this question will not even be "yes" for arbitrary non-commutative integral domains.