Left annihilators characterized by GPIs

Abstract:

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