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The power graph P G of a finite group G is the graph whose vertex set is G, two distinct vertices are adjacent if one is a power of the other. The order supergraph S G of P G is the graph with vertex set G, and two distinct vertices x, y are adjacent if o(x)|o(y) or o(y)|o(x). In this paper, we study the independence number of S G and answer a question was posed by Hamzeh and Ashrafi. [ABSTRACT FROM AUTHOR]

We investigate the formal matrix ring over R defined by a certain system of factors. We give a method for constructing formal matrix rings from non-negative integer matrices. We also show that the principal factor matrix of a binary system of factors determine the structure of the system. [ABSTRACT FROM AUTHOR]

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]