Answer to Question #23264 in Abstract Algebra for Melvin Henriksen

Question

Show that every nonzero homomorphic image of R= End(Vk) where V is a vector space over a division ring k is a prime ring.

Answer 1

For given ring it is known that allideals of R are linearly ordered by inclusion and idempotent, and thusthey are prime. If I is prime ideal of R then for any ideals AB in I we have that
A is in I or B is in I. Thus image of I will be 0 in R/I and images of A and B
will be some ideals A', B', The inclusion A'B' is in 0 is preserved, and its
conclusion A' is 0 or B' is 0 is preserved too. Thus R /I is prime ring.