Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice

Let ℜ be the finite commutative ring with unity and <i>I<sub>s</sub></i> be the S-prime ideal of a ring ℜ. The set <i>L<sub>s</sub></i> forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (<i>L<sub>s</sub></i>, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end.