Some new class of ideals in semirings and their applications

In the present paper, we focus on semirings, which are positive cones of a class of lattice-ordered rings. We establish a lattice isomorphism between semiring l-ideals and ring l-ideals of cancellative l-semirings and its difference l-ring, and from this, we obtain a structure of semiring l-ideals via ring l-ideals in cancellative l-semirings. Smith in [26] defined f-semirings as a suitable class of l-semirings in which one can establish a structure theorem. A new class of f-semirings, namely P-semirings, is defined to focus solely on positive cones of abundant function rings, e.g., C(X). We bring notions of z-ideals and z?-ideals into commutative semirings. It is shown that these ideals are equally important in investigating P-semirings like C+ (X). The structure of z-ideals and z?-ideals are obtained in P-semirings via the z-ideals and z?-ideals of its difference ring. We show that each k-ideal of a P-semiring is a z-ideal if and only if it is a von Neumann regular semiring.