On γ-Sets in Rings

Authors

  • Eva Jenny C. Sigasig
  • Cristoper John S. Rosero
  • Michael Jr. Patula Baldado Negros Oriental State University

DOI:

https://doi.org/10.29020/nybg.ejpam.v14i1.3873

Keywords:

$\gamma$-set, separating $\gamma$-set, ring, $c$-number

Abstract

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

Author Biography

Michael Jr. Patula Baldado, Negros Oriental State University

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How to Cite

Sigasig, E. J. C., Rosero, C. J. S., & Baldado, M. J. P. (2021). On γ-Sets in Rings. European Journal of Pure and Applied Mathematics, 14(1), 314–326. https://doi.org/10.29020/nybg.ejpam.v14i1.3873