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
If R ∼ Mn(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 ∼ Mn(k).
Week 5 Assignment 1 Review: Nicely defined dynamic stack. The push and pop functions seem to work great. Good job Week 5 Assignment 2 Review: Program coded to spec and ran fine. Good definition of the InventoryBin class with a full set of accessor functions. To make the class reusable for other applications, you should have also defined a set of mutator functions. Good definition of the 3 utility functions. This was a good exercise of the stack abstract data type – you’re clearly doing Object Oriented Programming now.