SCHWEITZER 1976 Problem 4

Let $\mathbb{Z}$ be the ring of rational integers. Construct an integral domain $I$ satisfying the following conditions:

a) $\mathbb{Z} \varsubsetneqq I$ ;

b) no element of $I - \mathbb{Z}$ (only in $I$ ) is algebraic over $\mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $\mathbb{Z}$ );

c) $I$ only has trivial endomorphisms.

E. Fried