作者author (AUTHOR)
看板Math
標題[代數] 幾題有關module的問題
時間Sat May 21 20:18:28 2011
1.
Prove that a ring R having the property that every finitely generated
R-module is free is either a field or the zero ring.
2.
A module is called simple if it is not the zero module and if it has no
proper submodule.
(a) Prove that any simple R-module is isomorphic to an R module of the
form R/M, where M is a maximal ideal.
(b) Prove Schur's lemma: let Φ: S→S be a homomophism of simple modules.
Then Φ is either zero, or an isomorphism.
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◆ From: 140.112.7.214
1F：推 Bourbaki :哈 是同班的厚 05/21 20:40
2F：→ Bourbaki :第一題考慮<a> a不為零則等於R maximal部分是trivial 05/21 20:41
3F：→ Bourbaki :field就是不會有nontrivial proper ideal 05/21 20:42
4F：→ Bourbaki :第二題的第一小題用correspondence Thm做 05/21 20:43
5F：推 Bourbaki :第二小題考慮ker等於{0}和等於S的狀況 然後再討論im 05/21 20:45
6F：推 Bourbaki :第二題的(a)因為S是simple 所以submodule <a>=R 05/21 20:52
7F：→ Bourbaki :做homo從R到S 1->a 在考慮ker 05/21 20:53