Exercise 1, § 2

Let A be a totally ordered ring and B an additive subgroup of A which is closed under multiplication. a) An element $x \in A$ is said to be infinitely large with respect to B if $|y| < |x|$ for all $y \in B$. Show that the set F(B) of elements of A which are not infinitely large with respect to B is a subring of A containing B. b) An element $x \in A$ is said to be infinitely small with respect to B if, for all $y > 0$ in B, we have $|x| < y$. If, for all $y > 0$ in B, there exists $z \in B$ such that $0 < z < y$, then the set of elements of A which are infinitely small with respect to B is a sub-pseudo-ring I(B) of A. If in addition, for every pair of elements y, z of B with $0 < y < z$, there exists $x \in B$ with $0 < xz < y$, then I(B) is an ideal in the ring F(B).