B. Please hand in the following additional problem.

In a commutative ring (R, +, ×), show that for all x ∈ R, (x) = x.

Assume that the integers, Z, satisfiy the axioms of a commutative ring. Show that the construction of the rational numbers (Q, +, ×), given on pp. 1718 satisfies the distributive axiom "D" on p. 16.

Suppose that the relation ≡ on the integers Z is given for x, y ∈ Z by
x ≡ y ⇔ ( ∃ z ∈ Z ) ( x = y + 7z ).
 Show that ≡ is an equivalence relation.
 Show: the binary operation given for equivalence classes â, û &isin Z/≡ by
â # û := (a + u)^{^}
is well defined. (That is, if we used different numbers in the equivalence classes b ≡ a and v ≡ u, then the answer is still the same equivalence class: a + u ≡ b + v.)
