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This work is motivated by the articles [9] and [19] in which the following two problems are solved. Let O be a finitely generated ℤ-algebra that is an integrally closed domain of characteristic zero, consider the following problems: (A) Fix s that is integral over O, describe all t such that O[s]= O[t]. (B) Fix s and t that are integral over O, describe all pairs (m, n) ϵ ℕ² such that O[sm]= O[tn]. In this article, we solve these problems and provide a uniform bound for a certain "discriminant form equation" that is closely related to Problem (A) when O has characteristic p > 0. While our general strategy roughly follows [9] and [19], many new delicate issues arise due to the presence of the Frobenius automorphism x → xp. Recent advances in unit equations over fields of positive characteristic together with classical results in characteristic zero play an important role in this article. [ABSTRACT FROM AUTHOR]