I have begun reading Herstein's book, "Noncommutative Rings". I like it very much in general, great style. So if someone reading this hasn't seen it yet, I think I can recommend it.
I'm writing this because I've having difficulty understanding a part of the text about Schur's lemma. I understand the statement and the proof since they're straightforward. What I'm having trouble with is the special case he's talking about later on. He says this:
I understand everything but the last sentence. Are the blocks supposed to be square matrices? Of the same size? What is the quantificator for andLet be a field and let be the ring of all matrices over We consider as the ring of all linear transformations on the vector space of tuples of elements of If is a subset of let be the subalgebra generated by over Clearly is a faithful and so, module. is, in addition, both a unitary and irreducible module.
We say the set of matrices is irreducible if is an irreducible module. In matrix terms this merely says that there is no invertible matrix in so that