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Let ℚ be the rational filed. For a square-free integer d with d ≠ 0 , 1 , we denote by K = ℚ (d) the quadratic field. Let K be the ring of algebraic integers of K. In this paper, we completely determine the unit group of the quotient ring K / 〈 ξ n 〉 of K for an arbitrary prime ξ in ℚ (d) , where ℚ (d) has the unique factorization property, and n > 1 is a rational integer. [ABSTRACT FROM AUTHOR]

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. Let R be an associative (not necessarily commutative) ring. We focus on the domination number of the zero-divisor graph Γ (R) , the compressed zero-divisor graph Γ E (R) and the unit graph G (R). We find some relations between the domination number of the zero-divisor graph and that of the compressed zero-divisor graph. Moreover, some relations between the domination number of Γ (R) and Γ (R [ x ; α , δ ]) , as well as the relations between the domination number of G (R) and G (R [ [ x ; α ] ]) , are studied. [ABSTRACT FROM AUTHOR]

We investigate the bilinear forms graph Γ over the residue class ring modulo p s (where p is a prime number and s is a positive integer). First, we prove that the bilinear forms graph Γ is a connected vertex transitive graph. When p > 2 , Γ is distance-regular if and only if s = 1 . Next, we completely determine the valency of a vertex, the clique number, the independence number and the chromatic number of the bilinear forms graph Γ, respectively. Finally, we show that both Γ and the complement of Γ are not cores and their cores are complete. [ABSTRACT FROM AUTHOR]

Ring extensions are a well-studied topic in ring theory. In this paper, we study the structure of the Gauss extension of a Galois ring. We determine the structures of the extension ring and its unit group. [ABSTRACT FROM AUTHOR]

The unit graph of a ring R, denoted G(R), is the simple graph defined on the elements of R with an edge between vertices x and y iff x + y is a unit of R. In this paper, we prove that the girth (G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. We determine the rings R with R/J(R) semipotent and with (G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with (G(R)) = 3 or 4. [ABSTRACT FROM AUTHOR]