Let $G$ be an affine algebraic group. A character of $G$ is a morphism $G\to \mathbb G_m$. Let $X$ be the abelian group of all characters of $G$. Suppose this group is finitely generated, say by ${\chi_1,\ldots, \chi_n}$, with no $p$-torsion, where $p$ is the characteristic of the field. We have the following theorem. Suppose $Y$ is a subgroup of $X$ such that $X/Y$ has no $p$-torsion. If $\chi$ is a character having the property that
$$ \bigcap_{\eta\in Y}\ker\eta\subset \ker\chi,$$
then $\chi\in Y$.
I think I have a counterexample to this theorem, and I am not able to see why it does not work. Suppose $Y = \langle \{\chi_1,\chi_2\}\rangle$, where $\ker\chi_1\cap\ker\chi_2 = \{e\}$. Then, $X/Y\cong \langle\{\chi_3,\ldots,\chi_n\}\rangle$ has no $p$-torsion. Moreover, any character will contain $\ker\chi_1\cap\ker\chi_2$, but they are obviously not all in $Y$.
Can you please help me with the proof of this theorem?
