The following generic ring functions are applicable to the ring of integers and its elements.
Create the abelian group of integers under addition. This returns an infinite (additive) abelian group A of rank 1 together with a map from A to the ring of integers Z, sending A.1 to 1.
Create the abelian group of invertible integers, that is, an abelian group isomorphic to the multiplicative subgroup < - 1 >. This returns an (additive) abelian group A of order 2 together with a map from A to the ring of integers Z, sending A.1 to -1.
The class group of the ring of Z (which is trivial).
Create the field of fractions Q of the ring of rational integers.
Given Z, the ring of integers or an ideal of it, and an element n of Z, create the ideal aZ∩Z of the ring of integers. Note that this creates an ideal, not just a subring.
The signature of Z as an order of Q, i.e. 1, 0.