Prime Ideal, Semiprime Ideal, and Radical of an Ideal of an L-Subring

In this paper, we develop a systematic theory for the ideals of an L-ring L(μ, R). We introduce the concepts of a prime ideal, a semiprime ideal, and the radical of an ideal in an L-ring. The notion of a maximal ideal has been introduced and discussed in different studies. We prove several results pertaining to these notions which are versions of their counterparts in classical ring theory. Besides this, we prove that for a commutative ring R, the radical η of an ideal η in an L-ring L(μ, R) is an ideal of μ provided that η has sup-property.