Let Z denote the integers Show that any subgroup of Z has th

Let Z denote the integers. Show that any subgroup of Z has the form nZ for some integer n.

Clear and detailed answer please

Let Z denote the integers. Show that any subgroup of Z has the form nZ for some integer n.

Clear and detailed answer please

Let Z denote the integers. Show that any subgroup of Z has the form nZ for some integer n.

Clear and detailed answer please

Proof :

The zero ideal is of the required form with n = 0.

Let I be some non-zero ideal of Z.

Then
I contains at least one strictly positive integer (since m I for all m I).

Let n be the smallest strictly positive integer belonging to I.

If j I then we can write j = kn + q for some integers k
and q with 0 q < n.

Now q I, since q = j kn, j I and kn I. But 0 q < n, and n is by denition the smallest strictly positive integer belonging to I.

We conclude therefore that q = 0, and thus j = kn.

This shows that I = nZ, as required.