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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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    Yes, we can say that they are isomorphic as vector spaces. But how about $\mathbb{C}$ and $\mathbb{R} \times \mathbb{R}$ as $\mathbb{R}$-algebras? They certainly aren't isomorphic as algebras.2012-01-10
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    @t.b.: your comment is correct; but in a question on Mackey's criteria on Mathoverflow, the first answer does a similar kind of work i.e, there, showing isomorphism of vector spaces will imply isomorphisms of group algebras in the first answer to question. (Link: http://mathoverflow.net/questions/84510/mackey-irreducibility-criteria)2012-01-10
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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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    This is the kind of question which you yourself could have answered by looking for examples...2012-01-10
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    Yes. But I didn't think abou it because of some arguments in the first answer to a question ("http://mathoverflow.net/questions/84510/mackey-irreducibility-criteria" )2012-01-10
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    In that context, Vladimir has a morphism of finite dimensional algebras $A\to B$. If you prive that it is an isomorphism of vector spaces, then it follows that it is an isomorphism of algebras. But you have to *start* with a morphism of algebras. (This is a classic example where explaining the motivation for your question would have allowed more useful answers...)2012-01-10
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    how this vector space isomorphism is sufficient for isomorphism of algebras? (I confused about it).2012-01-10
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    Well, suppose that $f:A\to B$ is an homomorphism of algebras *and* an isomorphism of vector spaces. Can you tell me what do you need to do to show that $f$ is an isomorphism of algebras? Can you do it?2012-01-10
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    that f is bijective?2012-01-10
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    Yes. Can you show that $f$ is bijective under those hypotheses?2012-01-10
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    Frobenius reciprocity has absolutely nothing to do with it, really.2012-01-10
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    What does the hypothesis that «$f$ is an isomorphism of vector spaces» mean?2012-01-10
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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