Let S = {a, b, c, ...} and ${Gamma}$ = {${alpha}$, ${eta}$, ${gamma}$, ...} be two nonempty sets. S is called a ${Gamma}$-semigroup if $a{alpha}b{in}S$, for all ${alpha}{in}{Gamma}$ and a, b ${in}$ S and $(a{alpha}b){eta}c=a{alpha}(b{eta}c)$, for all a, b, c ${in}$ S and for all ${alpha}$, ${eta}$ ${in}$ ${Gamma}$. An element $e{in}S$ is said to be an ${alpha}$-idempotent for some ${alpha}{in}{Gamma}$ if $e{alpha}e$ = e. A ${Gamma}$-semigroup S is called an E-inversive ${Gamma}$-semigroup if for each $a{in}S$ there exist $x{in}S$ and ${alpha}{in}{Gamma}$ such that a${alpha}$x is a ${eta}$-idempotent for some ${eta}{in}{Gamma}$. A ${Gamma}$-semigroup is called a right E-${Gamma}$-semigroup if for each ${alpha}$-idempotent e and ${eta}$-idempotent f, $e{alpha}$ is a ${eta}$-idempotent. In this paper we investigate different properties of E-inversive ${Gamma}$-semigroup and right E-${Gamma}$-semigroup.