The Source of Semi-Primeness of Γ-Rings

Abstract

The notion of source of semi-primeness is firstly given by Aydın, Demir and Camcı in 2018 as the set of all elements a of R that satisfy aRa 􀀀 0 for any associative ring R. They investigated some basic properties of this set and defined three types of rings which have not appeared in literature before. The theory of gamma ring has been introduced by Nobusawa in 1964 as a generalization of rings. In this work, we generalized the notion of source of semi-primeness for gamma rings and investigated its basic algebraic properties. We also defined SSMS -strongly reduced Γ-ring, SSMS -domain, SSMS -division ring and examined the relationship between these structures. We determined all possible characteristic values of a SSMS -domain and proved every finite SSMS -domain Γ-ring M is a SSMS -division Γ-ring.

Keywords

• $\Gamma$-ring
• source of semi-primeness
• strong unity

References

1. Aydın N., Demir C., Camcı D.K., The source of semiprimeness of rings, Communications of the Korean Mathematical Society, 33(4), 1083-1096, 2018.
2. Barnes W., On the Γ -rings of Nobusawa, Pacific Journal of Mathematics, 18(3), 411-422, 1966.
3. Coppage W.E., Luh J., Radicals of gamma rings, Journal of the Mathematical Society of Japan, 23(1), 40-52, 1971.
4. Kandamar H., The k-derivation of a Gamma-ring, Turkish Journal of Mathematics, 23(3), 221-229, 2000.
5. Kyuno S., On the radicals of Γ-rings, Osaka Journal of Mathematics, 12(3), 639-645, 1975.
6. Kyuno S., On prime gamma rings, Pacific Journal of Mathematics, 75(1), 185-190, 1978.
7. Kyuno S., Gamma Rings, Hadronic Press, Inc., 1991.
8. Kyuno S., A gamma ring with minimum conditions, Tsukuba Journal of Mathematics, 5(1), 47-65, 1981.

9. Luh J., On the theory of simple Γ-rings, The Michigan Mathematical Journal, 16(1), 65-75, 1969.
10. McCoy N.H., The Theory of Rings, MacMillan, 1968.
11. Nobusawa N., On a generalization of the ring theory, Osaka Journal of Mathematics, 1(1), 81-89, 1964.