Subrings in Quadratic Fields Which are Not Universal for GE2

Let F be a quadratic field and let S be a finite set of places containing S∞, the set of infinite places in F. Let v be a finite place outside S, S′ = S ∪ {v}, R = OS, and R′ = OS′. In this paper, some examples are given to show that if the prime ideal P in R corresponding to v is non‐principal or the natural homomorphism R^* → (R/P)^* is not surjective then, although R is universal for GE2, R′ may not be universal for GE2.