Answer to Question #17111 in Abstract Algebra for Tsit Lam

Question

R be a simple ring that is finite-dimensional over its center k, show that R is isomorphic to a matrix
algebra over its center k iff R has a nonzero left ideal A with (dimkA)2 ≤ dimkR

Answer 1

If R ∼ M_n(k), we can take A to be R · E11 for which (dimkA)² = n²= dimkR. Conversely, suppose R has a nonzero left ideal A with(dimkA)² ≤ dimkR. Then (nd)²= (dimkV )² ≤ (dimkA)² ≤ dimkR= n²d. This implies that d = 1, so D =k and R ∼ M_n(k).