Book recommendation for non-commutative algebra

I'm relatively new to representation theory and want to read up on the very basics. Most introductory representation theory books actually cover the theory of algebras over a field, but I'd like to read more general results from non-commutative algebra, i.e. over non-commutative rings and associative algebras. During my online search, it seems that Lam's First Course is the canonical recommendation, but I find it incredibly hard to read (up to sec. 3). Some more elementary facts are just assumed (homomorphisms of finite direct products can be written as matrices, although Lam blackboxes this to linear algebra over division rings??), some I think important parts are not covered (general isotypic decompositions are only a small exercise, and a quick search almost never mentions them), they always go on a complete side tangent at the end of sections (e.g. a lot of theory on 2×2-matrices in sec. 1, twisted and differential polynomial rings in sec. 3). But I must say the exercises are quite good. Is there another well-written, well-motivated yet comprehensive book on that matter? I'm thinking of books similar to Atiyah-MacDonald or even Matsumura's Commutative Ring Theory. Or even Stein-Shakarchi's Complex Analysis.