Abstract Let and be distinct primes, and let be an abelian pro‐‐group. We study the structure of the algebra and of ‐modules. The algebra turns out to be a direct product of copies of rings of integers of unramified cyclotomic extensions of , and this induces a similar decomposition for a family of ‐modules. Inside this family we define Sinnott modules and provide characteristic ideals and formulas à la Iwasawa for orders and ranks of their quotients. When is the Galois group of an extension of global fields, ‐class groups and (duals of) ‐Selmer groups provide examples of Sinnott modules, and our formulas vastly extend results of Washington and Sinnott on ‐class groups in ‐extensions. Moreover, for global function fields of positive characteristic, we use the specialization of a Stickelberger series to define an element in which interpolates special values of Artin ‐functions. With this element and the characteristic ideal of ‐class groups, we formulate an Iwasawa main conjecture (IMC) for this setting and prove some special cases of it for relevant ‐extensions.
Bandini et al. (Mon,) studied this question.