On finitely embedded rings

A result of Ginn and Moss asserts that a left and right noetherian ring wit h essential right socle is left and right artinian. There are examples of r ight finitely embedded rings with ACC on left and right annihilators which are not artinian. Motivated by this, it was shown by Faith that a commutati ve, finitely embedded ring with ACC on annihilators (and square-free socle) is artinian (quasi-Frobenius). A ring R is called right minsymmetric if, w henever kR is a simple right ideal of R, then Rk: is also simple. In this p aper we show that a right noetherian right minsymmetric ring with essential right socle is right artinian. As a consequence we show that a ring is qua si-Frobenius if and only if it is a right and left mininjective, right fini tely embedded ring with ACC on right annihilators. This extends the known w ork in the artinian case, and also extends Faith's result to the non-commut ative case.