Concretely the profinite integers will be the set of maps such that and . Pointwise addition and multiplication makes it a (non-integral) commutative ring. If a sequence of integers converges modulo n for every n then the limit will exist as a profinite integer.
A usual (rational) integer is a profinite integer since there is the canonical injection
There is a canonical pairing
where is the character of induced by . The pairing identifies with the Pontryagin dual of .
- Milne, Ch. I Example A. 5.
- Questions on some maps involving rings of finite adeles and their unit groups.
- Connes–Consani, § 2.4.
- K. Conrad, The character group of Q