Idea Definition(Ri Given a ring R and a nonempty set I CR. I is an ideal of R if it subjects to For any a,b∈I,a-b∈1. For any a∈I,r∈R,ar,Ta∈I. Definition( Lattice) A subset of a lattice is an ideal if it is a sublattice of L and x∈ I and a∈ L imply that a na∈I A proper ideal I of L is prime if a,b∈ L and a∩b∈I imply that a∈Iorb∈rIdeal Definition (Ring) Given a ring R and a nonempty set I ⊆ R. I is an ideal of R if it subjects to: 1 For any a, b ∈ I, a − b ∈ I. 2 For any a ∈ I, r ∈ R, ar, ra ∈ I. Definition (Lattice) A subset I of a lattice L is an ideal if it is a sublattice of L and x ∈ I and a ∈ L imply that x ∩ a ∈ I. A proper ideal I of L is prime if a, b ∈ L and a ∩ b ∈ I imply that a ∈ I or b ∈ I. Yi Li (Fudan University) Discrete Mathematics March 6, 2012 4 / 1