(untitled) (European Journal of Pure and Applied Mathematics, 2021)
Let R be a ring with identity 1R. A subset J of R is called a Î³-set if for every a âˆˆ R, there exist b, c âˆˆ J such that a+b = 0 and ac = 1R = ca. A Î³-set of minimum cardinality is called a minimum Î³-set. In this study, we identified some elements of R that are necessarily in a Î³-sets, and we presented a method of constructing a new Î³-set. Moreover, we gave: necessary and sufficient conditions for rings to have a unique Î³-set; an upper bound for the total number of minimum Î³-sets in a division ring; a lower bound for the total number of minimum Î³-sets in a division ring; necessary and sufficient conditions for T(x) and T to be equal; necessary and sufficient conditions for a ring to have a trivial Î³-set; necessary and sufficient conditions for an image of a Î³-set to be a Î³-set also; necessary and sufficient conditions for a ring to have a trivial Î³-set; and, necessary and sufficient conditions for the families of Î³-sets of two division rings to be isomorphic.