This paper concerns product
integrals of functions with values in a normed complete ring. The inverses of elements
obtained as such integrals are investigated. In particular, the conditions under which
[x∏y(1 + G)]−1 exists are shown to be related to the requirement that ∫xy|G2| = 0.
Since the existence of [x∏y(1 + G)]−1 is connected with the existence of the
product integrals y∏x(1 + G) and x∏y(1 − G), the study of the inverse
leads to a study of the conditions under which these integrals exist when
x∏y(1 + G) is known to exist. Commutative and noncommutative rings are
considered.
