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Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
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Linear Algebra Texts?
I rather like Linear Algebra Done Right, and depending on the type of students you are aiming the course for, I would recommend it over Hoffman and Kunze. Since you seemed worried that Axler might be …