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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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No, the cardinalities alone are not enough to identify a local commutative ring. For example, you can take $R = \mathbb F_p[x]/\langle x^3 \rangle$ and $S = \mathbb F_p[x,y] / \langle x^2, xy, y^2\ran …
answered Oct 19 '12 by Dustin Cartwright