good introduction to algebra over a field?

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The textbooks on linear algebra that I have read generally don't treat algebras over fields.

Are there any good textbooks that are the "sequels" to a typical linear algebra course, which treat algebras over fields?

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The book An Introduction to Nonassociative Algebras by Richard D. Schafer is a well known book on finite-dimensional algebras over a field. It is written in a plain and clear style, and it reads very smoothly.