Answer to Question #23262 in Abstract Algebra for Melvin Henriksen

Question

Let R = End(Vk) where V is a vector space over a division ring k. Show that all ideals not equal R are prime.

Answer 1

It is sufficient to show that allideals of R are linearly ordered by inclusion and idempotent. If dimkV <∞, R is a simple ring. Therefore, it suffices to treat thecase when V is infinite-dimensional. The ideals of R are linearlyordered by inclusion. To show that they are all idempotent, consider any ideal
nonzero I. There exists an infinite cardinal β <dimk V such that I = {f ∈ R : dimk f(V ) < β}. For any f ∈ I, let f' ∈ R be such that f' is the identity on the f(V), and zero on a direct complement of f(V ). Clearly, f' ∈ I, and f = f'f. Therefore, f ∈ I^2, and we have proved that I = I^2.