Abstract: | For a square-free integer d other than 0 and 1, let K=?(d), where ? is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over ?. For several quadratic fields K=?(d), the ring Rdof integers of K is not a unique-factorization domain. For d<0, there exist only a finite number of complex quadratic fields, whose ring Rd of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = ?1,?2,?3,?7,?11,?19,?43,?67,?163. Let ? denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/?vn? was determined by Cross in 1983 for the case d = ?1. This paper completely determined the unit groups of Rd/?vn? for the cases d = ?2,?3. |