Left annihilators characterized by GPIs

Abstract

Let R R be a semiprime ring with extended centroid C C , U U the right Utumi quotient ring of R R , S S a subring of U U containing R R and ρ 1 {\rho _1} , ρ 2 {\rho _2} two right ideals of R R . In the paper we show that l S ( ρ 1 ) = l S ( ρ 2 ) {l_S}({\rho _1}) = {l_S}({\rho _2}) if and only if ρ 1 {\rho _1} and ρ 2 {\rho _2} satisfy the same generalized polynomial identities (GPIs) with coefficients in S C SC , where l S ( ρ i ) {l_S}({\rho _i}) denotes the left annihilator of ρ i {\rho _i} in S S . As a consequence of the result, if ρ \rho is a right ideal of R R such that l R ( ρ ) = 0 {l_R}(\rho ) = 0 , then ρ \rho and U U satisfy the same GPIs with coefficients in the two-sided Utumi quotient ring of R R .