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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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Subsets of the integers which are closed under multiplication
To make Mark Sapir's answer more explicit, we can take the direct product of sets obtained from your second example. For instance, we can take $S = \{ x: \exists a \in A, b \in B: x =ab\}$ where $A = …