# integral extension and existence of finitely generated faithful module

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 $$B=$$ 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 $$u\in B'$$.