I'm looking for a good algebra book that covers algebras. We are currently using Rotman's Advanced Modern Algebra for my algebra class, but unfortunately it only covers algebras as a subtopic of modules. In class we are covering algebras as a separate topic, and we haven't done module theory yet. The lecturer is working off his own notes for algebras, and they are, let's say, less than clear.
The topics we have covered / will cover are basically Frobenius' theorem for classifying the finite-dimensional division algebras over $\mathbb R$ and a lot of smaller results about the nilradical and prime radical.
I would like a recommendation for a book that covers algebras as a separate topic, preferably without basing it on module theory, that I can use to revise for my preliminary exams at the end of the summer.
EDIT: This is the proof of Frobenius' theorem from our notes, to give you an idea of how we are working with algebras:
Topics in Algebra by Herstein discusses Frobenius' theorem in chapter 7.
A highly-recommended standard reference for algebras is Associative Algebras by Pierce, but this book probably has much more material than you need or want. Frobenius' theorem appears in chapter 13.