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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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Does base extension reflect the property of being isomorphic?
I hope I'm not misunderstanding the question. Here goes:
We'll show that if $M,N$ are finite-dimensional over $K$,
then they are isomorphic over $K$.
Think of the linear space $X=\mathrm{Hom}_{A}( …