# Answer to Question #17111 in Abstract Algebra for Tsit Lam

Question #17111

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

algebra over its center k iff R has a nonzero left ideal A with (dimkA)2 ≤ dimkR

Expert's answer

If

*R**∼*M*(*_{n}*k*), we can take A to be*R · E*11 for which (dim*k*A)^{2}=*n*^{2}= dim*kR.*Conversely, suppose*R*has a nonzero left ideal A with(dim*k*A)^{2}*≤*dim*kR*. Then (*nd*)^{2}= (dim*kV*)^{2}*≤*(dim*k*A)^{2}*≤*dim*kR*=*n*^{2}*d.*This implies that*d*= 1, so*D*=*k*and*R**∼*M*(*_{n}*k*).
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