The importance of the concept of primitive ideals of associative rings consists in the well-known theorem stating that every semisimple ring A is a subdirect sum of primitive rings B, where a ring A is called semisimple (in the sense of Jacobson) if the Jacobson radical, i.e. the intersection of all primitive ideals, coincides with the zero ideal (0), and a ring B, is called primitive if the ideal (0) is a primitive ideal of B,. (Cf. N. Jacobson, Structure of rings, Colloq. Publ., Vol. 37, Amer. Math. Soc., Providence, R. I., 1956.) Some new characterizations were recently given for the Jacobson radical of a ring A. For instance, A. Kertesz [3] has shown in these Proceedings (generalizing an observation of L. Fuchs [1]) that the Jacobson radical J of a ring A consists of exactly those elements x of A for which the product yx lies with every yeA in the Frattini Asubmodule of the ring A, as of an A -right module A for itself (cf. also Hille [2]). Furthermore A. Kertesz [4] has shown that J is the intersection of all those maximal right ideals R of A for which there must exist, for any element xElR (xCA), a second element yCA with yxE:R; that is, those right ideals for which A-1RCR holds, where X-'R= {y; yEA, XyCR} for an arbitrary subset X of A. Furthermore, let LX Y-1 denote the subset { z; zEA, z YCL }. Every modular right ideal R of A is quasi-modular in the sense that A-1RCR holds. The concept of quasi-modularity of riglit ideals R was introduced in [6]. Solving a problem proposed by Kertesz [4] I have shown in [6] the existence of an associative ring which has a quasi-modular maximal but not a modular right ideal. In my other paper [7] a two-sided ideal Q of A is called quasi-primitive if there exists a quasi-modular maximal right ideal R of A with Q=A-lRCR. Obviously every primitive ideal is also quasi-modular in A, and almost trivially every artin ring with (0) quasi-primitive ideal is a total matrix ring over a skew field. Furthermore, any quasi-primitive ideal is clearly a prime ideal, and any commutative ring with (0) quasi-primitive ideal is a field. Solving a problem of my colleague Dr. Steinfeld, I have proved in [7] that the Jacobson radical J of A must coincide with the intersec-