Let a and b be elements of a ring such that a, b, and ab - 1 are units. The document shows that: 1) a-b-1 and (a - b-1)- a-1 are also units. 2) The identity ((a-b-1)-1 - a-1)-1 = aba - a holds. 3) It proves this identity by showing that each step results in a unit, and the final expression aba - a is a unit.

1. Let a, b be elements of a ring such that a, b and ab - 1 are units. Show that a-b -1
and (a - b -1
) - a -
1
are units, and the following identity holds: ((a-b -1
) -1
- a -1
) -1
= aba - a.
Solution
((a-b -1
) -1
-a -1
) -1
= (ab-1)a
ab-1 is unit then (ab-1)a = 1.a = a
((a-b -1
) -1
-a -1
) -1
=a
here a is a unit then
((a-b -1 ) -1 -a -1 ) -1 = 1 = unit
(a-b -1 ) -1 -a -1 = unit
(a-b -1
) -1 - a -1 = unit
take aba-a = unit
(ab-1) a = unit
ab-a = unit
b(a-ab -1
) = unit
a-ab -1
= unit
then ab -1
=b -1
since a is unit
a- b -1
= unit