Let R be a ring with nonzero identity. The unit graph of R, denoted by G R , has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G R are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G R are given. (These terms are defined in Definitions and Remarks 4.1, 5.1, 5.3, 5.9, and 5.13.)

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In this paper, we define certain equivalence relations called *-relations on ternary semigroups and we mention some properties of these relations. We study these relations in respect to Green’s relations in ternary semigroups and by bringing some examples, we show that while some propositions are correct for Green’s relations, they are not necessarily true for these relations. Then we investigate *-relations in certain ternary semigroups.

Abstract. Let C.R/ denote the center of a ring R and g.x/ be a polynomial of ring C.R/OEx. An element r 2 R is called "g.x/-clean" if r D s C u where g.s/ D 0 and u is a unit of R and R is g.x/-clean if every element is g.x/-clean. In this paper, we introduce the concept of g.x/-full clean rings and study various properties of them.