Does This Ring have a Name?

Question

Let $M_1=\{0,1,2,4,5,8,9,10,\cdots\}$ be the set of nonnegative integers that can be written as a sum of two perfect squares. Let $M_2=\{\sqrt{m}: m\in M_1\}=\{0,1,\sqrt{2},2,\sqrt{5},\cdots\}$. Let $R$ be the set of all real numbers that can be expressed as a finite linear combination of elements of $M_2$ with integer coefficients. So, for example, $3+2\sqrt{5}-6\sqrt{10}\in R$. Then $R$ forms a ring with the usual addition and multiplication defined on the reals. Does $R$ have a specific name? Does it appear anywhere in the literature or is there anything at all that I can read about it? Thank you.