A generalization of (FI-)extending and (quasi) Baer rings

A ring [Formula: see text] is called right essentially (quasi) Baer if the right annihilator of every (ideal) nonempty subset of [Formula: see text] is essential in a right ideal generated by an idempotent. This class of rings includes both (FI-) extending and (quasi) Baer rings. Initially we develop basic properties and give some examples of these rings. Later the right essentially quasi-Baer property is shown to carry over product, matrix, monoid, polynomial and upper triangular matrix rings.