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$\begingroup$

The set

$$S= \left\{ \left(\begin{array}{cc}a&b \\ -b&a \end{array}\right):a, b\in\mathbb{R} \right\}$$

is a subring of the matrix ring $M_2(\mathbb R)$ isomorphic to $\mathbb C$. Can we find other subring $L$ of $M_2(\mathbb R)$ which is also isomorphic to $\mathbb C$ and such that $S\cap L= \left\{ \left(\begin{array}{cc}a&0 \\ 0&a \end{array}\right):a \in\mathbb R\right\}$?

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    As a hint, this is very easy to do. Any automorphism of M2R will give you a different C.2012-10-09
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    @zacarias: Sorry for sticking to details, but saying that some object is a subfield, implicitly assumes that the bigger object is a field. Probably you should say "subgroup which is a field"...2012-10-09
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    @DennisGulko, the term *subfield* does not carry the implication that the big thing is a field, in my experience —and I see the term quite often!2012-10-09
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    @Mariano Suárez-Alvarez: I see it quite often too. I guess it depends on the general context :)2012-10-09
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    For example, there are whole volumes written about subfields of central simple algebras, and I doubt anyone has ever written «subalgebras which are fields» to refer to them :-)2012-10-09

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