[代數] 2題有關module的問題
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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