For a square-free integer d other than 0 and 1, let K = Q（√d）, where Q is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over Q. For several quadratic fields K = Q（√d）, the r...For a square-free integer d other than 0 and 1, let K = Q（√d）, where Q is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over Q. For several quadratic fields K = Q（√d）, the ring Rd 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 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 Q denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/（Q^n） was determined by Cross in 1983 for the case d = -1. This paper completely determined the unit groups of Rd/（Q^n） for the cases d = -2, -3.展开更多
The unit graph of a ring is the simple graph whose vertices are the elements of the ring and where two distinct vertices are adjacent if and only if their sum is a unit of the ring.A simple graph is said to be planar ...The unit graph of a ring is the simple graph whose vertices are the elements of the ring and where two distinct vertices are adjacent if and only if their sum is a unit of the ring.A simple graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints.In this note,we completely characterize the semipotent rings whose unit graphs are planar.As a consequence,we list all semilocal rings with planar unit graphs.展开更多
基金This work was supported by the National Natural Science Foundation of China （Grant Nos. 11461010, 11161006）, the Guangxi Natural Science Foundation （2014GXNSFAAll8005, 2015GXNSFAA139009）, the Guangxi Science Research and Technology Development Project （1599005-2-13）, and the Science Research Fund of Guangxi Education Department （KY2015ZD075）.
文摘For a square-free integer d other than 0 and 1, let K = Q（√d）, where Q is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over Q. For several quadratic fields K = Q（√d）, the ring Rd 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 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 Q denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/（Q^n） was determined by Cross in 1983 for the case d = -1. This paper completely determined the unit groups of Rd/（Q^n） for the cases d = -2, -3.
基金supported by the National Natural Science Foundation of China(11661013,11661014,11961050)Guangxi Natural Science Foundation(2016GXNSFCA380014,2016GXNSFDA380017).
文摘The unit graph of a ring is the simple graph whose vertices are the elements of the ring and where two distinct vertices are adjacent if and only if their sum is a unit of the ring.A simple graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints.In this note,we completely characterize the semipotent rings whose unit graphs are planar.As a consequence,we list all semilocal rings with planar unit graphs.
