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If $A_1$ and $A_2$ are algebras over a field $F$, and if they are isomorphic as vector space over $F$, can we say that these algebras are isomorphic?

(one may assume that algebras are finite dimensional, if necessary; I don't know about it. But I am just wondering about vector space isomorphism implies algebra isomorphism. )

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    It does not *seem* to be correct: it *is* correct.2012-01-10

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No. As soon as $n$ is large, there are many, many non-isomorphic algebras of dimension $n$.

For example, the algebras $F\times F$ and $F[x]/(x^2)$ are isomorphic as vector spaces (they both have dimension $2$) but they are not isomorphic as algebras.

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    Actually, $f\colon A\rightarrow B$ is homomorphism of algebras and an isomorphism of vector space (which means it is bijective). Is it correct?2012-01-10