Could you explain why $I_{\beta}^{(m)}=I_{\gamma}^{(m)}$ for all $m$ implies $I_{\beta}=I_{\gamma}$?

Thanks. I would appreciate if you would add an explanation why $I_{\beta}$ is maximal.

By the way, Chinese and Japanese languages are very different. The difference is far bigger than that between Latin and English. Just because they both use Kanji does not change this fact.

Almost all the Japanese mathematicians write papers in English, so you don't need to learn Japanese to read them. Some mathematicians write textbooks in Japanese, but if they are important, they are likely to be translated in English. For example, Matsumura's Commutative Ring Theory is such a textbook. So again you don't need to learn Japanese to read them.