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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
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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 = …