We find the smallest field containing the number 1 when using the usual addition and multiplication of real numbers.
We show that the integers modulo any prime forms an integral domain under addition and multiplication modulo p.
We show, step-by-step, that the set of all 2 x 2 matrices with real entries forms a ring under addition and multiplication.
We show that every cyclic group is isomorphic to either the set of integers or the set of integers mod some integer under addition.
We show that the composition of two isomorphisms is again an isomorphism.