Let R be a ring with the property that every element is either nilpotent or invertible. If a, b, c are in R with a and b nilpotent, show that ac, ca, and a + b are nilpotent. For the latter, first observe that a + b cannot equal 1. Conclude that Nil (R) is the set of all nilpotent elements of R.
(nil radical Nil (R) is defined to be the sum of all nil two-sided ideals of R)
Since a is nilpotent, we can find a positive integer k, such that a^k=0. We can assume that k is the smallest positive integer such that a^k=0.
Now I claim that ac is also nilpotent. If not, according to the condition, ac must be invertible. Then we can find some u in R, such that (ac)u=acu=1. But we note
a^(k-1) * acu=a^(k-1) = a^k * cu = 0*cu=0
So a^(k-1)=0. ...
This solution is comprised of a detailed explanation to show that ac, ca, and a + b are nilpotent.