Endomorphism rings of bimodules

Let $$M$$M be an $$R$$R-$$R$$R-bimodule over a semi-prime right and left Goldie ring $$R$$R. We investigate how non-singularity conditions on $$M_R$$MR are related to such conditions on $$_RM$$RM. In particular, we say an $$R$$R-$$R$$R-bimodule $$M$$M such that $$_RM$$RM and $$M_R$$MR are non-singular has the right essentiality property if $$IM_R$$IMR is essential in $$M_R$$MR for all essential right ideals $$I$$I of $$R$$R, and investigate several questions related to this property.