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Let $A$ be a commutative $\mathbb R$-algebra (with or without unit) and of finite dimension (when considered as a $\mathbb R$-vector space).

  1. Is there a structure theorem for such type of algebras or some classification of them?

  2. What about deformation theory for such algebras?

Any references are very much appreciated.

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    Even with unit, these are too broad. By Chinese remainder theorem, you can say that these are products of local rings. But, the classification of local ones does not exist. For question 2, these deformation spaces are called Hilbert schemes, but can be very complicated. – 2017-02-21
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    @Mohan What about partial results or results in small dimension? – 2017-02-21
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    What does small mean? There are probably results for dimension at most six or seven. – 2017-02-21
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    @Mohan yes, in this article http://math.mit.edu/~poonen/papers/dimension6.pdf the case of $\mathbb C$-algebras is treated for dimension $\leq 6$. But unfortunately there are no references to partial results for the case of $\mathbb R$-algebras – 2017-02-21
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    There is no difference between the two, thanks to Hensel's lemma. – 2017-02-21

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