Equal additive order of all elem in simple ring

[UD1]

Homework Statement

Let S be a simple ring. Show that all nonzero elements of S have equal additive order. Show that this order either is a prime number p or is infinite.

The Attempt at a Solution

All I could show is that the order of any element x in S must divide that of the unity element: if n is the order of the unity element, then nx=n(1x)=(n1)x=0x=0. By Lagrange's theorem the order of any element must also divide the number of elements in the ring.

 

[lippyka]

Since S is a simple ring, it has only two elements, namely 0 and 1, so the order of any element must divide 2. Thus, the order of any element is either 1, 2 (which is not allowed since all nonzero elements must have order greater than 1) or infinite.