Thread: Inverse

Define an operation * on Z by

a*b = ab - a - b +2 for a, b in Z. We know that * is commutative and associative and that * has an identity element, namely 2. Determine which elements in Z have inverses relative to *. Justify your answer.

Thanks very much

Originally Posted by suedenation

Define an operation * on Z by

a*b = ab - a - b +2 for a, b in Z. We know that * is commutative and associative and that * has an identity element, namely 2. Determine which elements in Z have inverses relative to *. Justify your answer.

Thanks very much

Given we attempt to find a such as,
because 2 is the identity element.
Thus,
by definition on .
Thus,
or more elegantly
Thus,

Now we examine 2 cases:
Case 1 : Thus, . Impossible.
Case 2 : Thus, . Thus if then Checking this we find that which is true. Thus, has an inverse. Now if also then,
with because they are right next to each other. Thus, the left hand side must be divisible by , thus, this is Euclid's Lemma. Thus, . Thus, now since thus, happens only when . Thus,
Thus, is the other inverse. Thus, are the only possible inverses.

It may help to note that (a+1)*(b+1) = (a+1)(b+1)-(a+1)-(b+1)+2 = ab+1. Thus the map (Z,*) -> (Z,.) given by a -> a+1 is an isomorphism of semigroups.