Unit groups of quotient rings of complex quadratic rings.

For a square-free integer d other than 0 and 1, let $$K = \mathbb{Q}\left( {\sqrt d } \right)$$, where $$\mathbb{Q}$$ is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over $$\mathbb{Q}$$. For several quadratic fields $$K = \mathbb{Q}\left( {\sqrt d } \right)$$, the ring R of integers of K is not a unique-factorization domain. For d < 0, there exist only a finite number of complex quadratic fields, whose ring R of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = −1,−2,−3,−7,−11,−19,−43,−67,−163. Let ϑ denote a prime element of R, and let n be an arbitrary positive integer. The unit groups of R/< ϑ> was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of R/< ϑ> for the cases d = −2,−3. [ABSTRACT FROM AUTHOR]