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I am reading the book Advanced modern algebra by Rotman. In the integral extension section, he claims that if $R^*/R$ is a ring extension and $u\in R^*$ be a nonzero element then

If there is a finitely generated $R$-submodule $B$ of $R^*$ with $uB\subset B$, then there exists a finitely generated faithful $R$-submodule $B'$ of $R^*$ with $uB'\subset B'$.

In the proof, he says if $B=<b_1,...,b_n>$ is a finitely generated $R-$submodule of $R^*$ with $uB\subset B$ then define $B'=<1,b_1,...,b_n>$ which is finitely generated and faithful.

But why $uB'\subset B'$? because it is not necessarily true that $u\in B'$.

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