Polynomial Arithmetic as described in this section was very interesting. I see how it relates to rings and the basic principles of arithmetic which have been covered so far. Yet the first thing I don’t understand comes from Theorem 4.1.
Theorem 4.1:
If R is a ring, then there exists a ring T containing an element x that is not in R and has these properties:

R is a subring of T

for every

…
Point 2 is a little confusing. Hungerford explains that this does not mean that the ring is commutative, except for x. How can there be rings that are non commutative in every way except for a certain value x which belongs to the ring? Unless he means that and for all . I realize this is what I typed above yet does Hungerford imply that or regardless of whether or not for every as long as .