Answer to Question #24417 in Algebra for Siavosh

Question

Show that :
1- If a module is integrally closed , then so its isomorphic copy .
2- Let M be a divisible projective R-module.Then R is integrally closed.

Answer 1

First statement is obvioussince isomorphism preserves all algebraic properties.
Second statement means that for any r in R we have M=rM and then M -projective
means that it is direct summand and thus rM is direct summand of free module
R^n, thus by criteria of integral element we have that R have to be intagrally
closed.