Abstract Algebra Question. Please answer a-e.

Question 2.15) Consider the set of rational numbers q that have the property that when q is written in the reduced form a/b with a, b integers and gcd(a, b) = 1 the denominator b is odd. This set is usually denoted by Z₍₂₎, and contains elements like 1/3, 5/7, 6/19, etc., but does not contain 1/4 or 5/62.

a) Does Z₍₂₎ contain 2/6?

Notice that every element of Z₍₂₎ is just a fraction (albeit of a particular kind). We know how to add and multiply two fractions together, so we can use this knowledge to add and multiply any two elements of Z₍₂₎. Here is the punch line: Z₍₂₎ forms a ring under the usual operation of addition and multiplication of fractions! Strange as this ring may seem at first, it plays an important role in number theory.

b) Check that if you add (or multiply) two fractions in Z₍₂₎ you get a fraction that is not an arbitrary rational number but one that also lives in Z₍₂₎. What role does the fact that the denominators are odd play in ensuring this? (The role of the odd denominators is rather crucial; make sure that you understand it!)

c) Why do associativity and distributivity follow from the fact that Z₍₂₎ Q?

d) Do the other ring axioms hold? Check!

e) Can you generalize this construction to other subsets of Q where the denominators have analogous properties?

(Clue & Remarks: Given two elements a and b in Z₍₂₎, write a as x/y where gcd(x, y) = 1 and y is odd. (Why can you do this?) Write b as u/v where gcd(u,v) = 1 and v is odd. Then a+b = (xv+yu)/yv. This fraction may not be reduced, but notice that yv, being a product of two odd integers, is odd. After you cancel all common factors from (xv + yu) and yv, the resultant fraction will still have an odd denominator (why?). Hence a + b will be in Z₍₂₎. In a similar way, show that ab (gotten by the usual multiplication of two rational numbers) will also be in Z₍₂₎. Now that you have two binary operations on Z₍₂₎, you can check that the ring axioms hold. As with previous examples, associativity and distributivity follow from the fact that they hold for the rationals. Notice that the fact that the product of two odd integers is odd was essential in showing that both a + b and ab lie in Z₍₂₎. How could we generalize this? Rewrite this property in the contrapositive form, yv is even implies either y or v is even, that is, 2|yv implies 2|y or 2|v. If we could find another integer n that has the property that n|yv implies n|y or n|v, we could use the same arguments to show that Z_(n) is also a ring. (Assuming you found such an integer n, how would you define Z_(n)?) Can you think of other integers that have this property?).