I’ve been trying to brush up on ring theory; it has been a long time since I studied rings in any depth and I need some ring theory to do some work in topology. In a previous post, I talked about ideal topologies and I might discuss divisor toplogies (starting with the ring of integers).
So, I grabbed an old text, skimmed the first part and came across an exercise:
an element is nilpotent if there is some positive integer
such that
. So, given
nilpotent in a commutative ring
one has to show that
is also nilpotent and that this result might not hold if
is not a commutative ring.
Examples: in the ring so
is nilpotent. In the matrix ring of 2 by 2 matrices,
and
are both nilpotent elements, though their sum:
is not; the square of this matrix is the identity matrix.
Immediately I thought to let be the smallest integers for
and thought to apply the binomial theorem to
(of course that is overkill; it is simpler to use
. Lets use
. I could easily see why
but why were the middle terms
also zero?
Then it dawned on me: for all
. Duh. Now it made sense. 🙂