 Dedekind also replaced Kummer’s “ideal  numbers” by his “ideals”—set-theoretically constructed objects intended to play the same role with respect to unique factorization. Dedekindian ideals are infinite subsets of the number fields in question, or of the ring of integers contained in them, which again gives his approach an infinitary character. (An ideal I in a ring R is a subset such that the sum and difference of any two elements of I and the product of any element of I with any element of R are also in I.) This led him to introduce other fruitful notions as well, such as that of a “module”. A particular  problem Dedekind struggled with for quite a while, in the sense of finding a fully satisfactory solution, was to specify a suitable version, not only of the notion of “integer”, but also of “prime number” (Ferreirós 1999, ch. 3, Avigad 2006, Haffner 2014).