Exercise 5, § 8

a) Let A be a semiprime ring such that every set of monogenous left ideals admits a minimal element. Prove that A is semisimple. (Using Exercise 7 of VIII, p. 130, construct sequences $(e_n)$ and $(f_n)$ of idempotents in A such that the ideals $Ae_n$ are minimal and that we have $Af_{n-1}= Ae_n\oplus Af_n$, and show that $f_n$ is zero for $n$ sufficiently large.)

b) Let K be a commutative field, V an infinite-dimensional right vector space over K, and A the K-subalgebra of End$_K(V)$ generated by $1_V$ and the endomorphisms of finite rank. Prove that every left ideal of A contains a minimal ideal.