In this paper, the notions of an L-sublattice, L-ideal, and L-dual ideal of an L-lattice are introduced. The concepts of an L-maximal ideal and an L-prime ideal in an L-lattice are also defined. The present paper also gives the precise structures of an L-sublattice, L-ideal (dual ideal) generated by an L-subset of an L-lattice. Finally, it is proved that under certain conditions, an L-maximal ideal in an L-lattice is L-prime. It is important to note that, in our studies, the evaluation lattice changes from $[0,1]$ to a lattice L and the parent structure shifts from an ordinary lattice to an L-lattice.

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 $\sqrt{\eta }$ of an ideal η in an L-ring L(μ, R) is an ideal of μ provided that η has sup-property.

In this paper, a systematic theory for the ideals of an L-ring $L(\mu ,R)$ has been developed. Earlier the authors have introduced the concepts of prime ideals, semiprime ideals, primary ideals, and radical of an ideal in an L-ring. They have also introduced and discussed the notion of a maximal ideal in different papers wherein several results pertaining to these notions have been proved. In this paper, the concepts of associated prime ideal, minimal prime ideal, and that of irreducibility of an ideal in an L-ring have been introduced, which in fact, are the continuation of authors’ previous works in the theory of ideals in L-rings.