Absolutely pure modules
Absolutely pure modules
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DOI：
10.1090/s0002-9939-1970-0294409-8
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发表时间：
1970-04
期刊：
影响因子：
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通讯作者：
C. Megibben
中科院分区：
文献类型：
--
作者：
C. Megibben
。证明了模A是绝对纯的当且仅当A上的每个有限相容线性方程组在A中都有解.根据这些环上的绝对纯模的性质刻画了Notherian环、半遗传环、正则环和Prüfer环.例如，R是Noether当且仅当每个绝对纯I？-模是内射和半遗传的当且仅当绝对纯I？-模类在同态映象下是闭的。如果R是Prüfer整环，则绝对纯I？-模是可除模，且当A可除且M是可数生成的无挠.R-模时，Ext\t(M，A)=0。
. A module A is shown to be absolutely pure if and only if every finite consistent system of linear equations over A has a solution in A. Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely pure modules over these rings. For example, R is Noetherian if and only if every absolutely pure i?-module is injective and semihereditary if and only if the class of absolutely pure i?-modules is closed under homomorphic images. If R is a Prüfer domain, then the absolutely pure i?-modules are the divisible modules and Ext\t(M, A)=0 whenever A is divisible and M is a countably generated torsion-free .R-module.