A Class of Ring-like Objects

We introduce the notions of one-sided dirings, 3-irreducible left modules, 3-primitive left dirings, 3-semi-primitive left dirings, 3-primitive ideals and 3-radicals. The main results consists of two parts. The first part establishes two external characterizations of a 3-semi-primitive left diring. The second part characterizes the 3-radical of a left diring by using 3-primitive ideals. By forgetting some structures of a 7-tuple introduced in Chapter 4 of [3], we get three roads of generalizing the notion of a ring R. The first one is to keep the additive group structure of R and to replace the multiplicative monoid structure of R by a dimonoid with a one-sided bar-unit. The second one is to replace the additive group structure of R by a commutative digroup and to keep the multiplicative monoid structure of R. The third one is to replace the additive group structure of R by a commutative digroup and to replace the multiplicative monoid structure of R by a dimonoid with a one-sided bar-unit. Although we do not know how far we can go along the third road now, the first two roads are good enough to develop the counterpart of the basic ring theory. The purpose of this paper is to study the counterpart of the Jacobson radical for rings along the first road. This paper consists of five sections. In Section 1 we introduce the notion of a one-sided diring and discusses its basic properties. In Section 2 we consider some fundamental concepts and results about a left module over a left diring. In Section 3 we introduce the notion of a 3-irreducible left module and prove that Schur Lemma is still true for 3-irreducible left modules over a left diring. In Section 4 we introduce the notions of 3-primitive left dirings and 3-semiprimitive left dirings, and establish two external characterizations of a 3-semiprimitive left diring. In Section 5 we introduce the notion of the 3-radical of a left diring by using the intersection of the annihilators of all 3-irreducible left R-modules, and prove that the 3-radical of a left diring R is equal to the intersection of the 3-primitive ideals of R.