In this paper, we impose some more conditions on a $\Gamma-$ semiring such as cancellative, centreless, semi subtractive, zero divisor, simple and characterize the results of a semiring by using cancellative semi- subtractive, zero divisor of cancellative semi- subtractive and cancellative rigid semi- subtractive $\Gamma-$ semiring with a strong identity. Furthermore, we study maximal and minimal ideals of strongly multiplicative $\Gamma-$ idempotent $\Gamma-$ semiring and the results regarding division $\Gamma-$ semiring, $\Gamma-$ semi field and $\Gamma-$ field for Izuka and Borne factor $\Gamma-$ semiring.
Keywords and Phrases
Multiplicatively cancellative $\Gamma-$ semiring, semi - subtractive, maximal and minimal ideals, division $\Gamma-$ semiring, Bourne and Izuka $\Gamma-$ congruence relation.
A.M.S. subject classiﬁcation
16Y60, 12E15, 46J20, 06B10, 12K10.