The Jacobson semiradical over a certain semiring

Abstract

The concept of the semiradical class of semirings was introduced in [3]. The purpose of this paper is to study one such semiradicals, the Jacobson semiradicals, over certain semirings. We generalize the concept of the Jacobson radical of a ring to a semiring. Some properties of the Jacobson semiradical $JS(R)$ of the semiring $R$ parallel those of ring theory. In Section 1 we describe some preliminary definitions. In Section 2 we define regular strongly austere semimodulse. Theorem 1 characterizes a regular strongly austere semimodule in terms of a regular modular maximal subtractive left ideal. We define $JS(R)$ and derive some properties of this structure. In Section 3 we show that the $JS(R)$ of a semiring has many representations as the intersection of left ideals. One of the more important of these is that $JS(R)$ is the intersection of all left weakly primitive subtractive ideals. Proposition 6 characterizes a semiweekly primitive semiring in terms of a weakly primitive semiring. The interrelationships of strongly austere, weakly primitive and semiweekly primitive semirings are examined in Theorem 3. In Section 4, Proposition 7 shows that the $JS(R)$ of a semiring with identity is the intersection of all regular maximal modular subtractive left ideals. Corollary 3 shows that $JS(R)$ is the unique largest superfluous left ideal of $R$. Proposition 8 shows that the class of Jacobson semiradical of semirings is closed under direct sum. We conclude with Section 5, a consideration of a certain restricted class of semirings. We show that the Jacobson semiradical for the semirings belonging to this class constitutes a semiradical class. Finally, Example 1 shows that a semiradical class need not be a radical class.