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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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Why is it a good idea to study a ring by studying its modules?
A simple example: If you have a representation of a group presented by generators and relations, you have an explicit algorithm to solve the word problem for this group. Another example: the proof of …