Ideallattice

Ideallattice, in mathematics, refers to the lattice formed by the collection of all ideals of a given algebraic structure, ordered by inclusion, with the lattice operations defined by intersection as the meet and the sum (the ideal generated by the union) as the join. The most common instance is the lattice of ideals of a ring.

For a ring R, the set Id(R) of all two-sided ideals forms a complete lattice, with bottom element 0 and top element R. The meet of any family of ideals is their intersection, and the join is the smallest ideal containing all of them, often written as the sum ∑ Iα. This lattice is a natural object of study in ring theory, algebraic geometry, and related areas, encapsulating how ideals nest and combine under addition and containment.

Examples help illustrate the structure. In the ring of integers Z, every ideal is of the form nZ for some n ≥ 0. The lattice Id(Z) is isomorphic to the divisibility lattice on natural numbers, with I ∨ J corresponding to gcd and I ∧ J to lcm (up to the usual units). In the polynomial ring k[x], which is a principal ideal domain, every ideal is principal, and the inclusion (f) ⊆ (g) holds precisely when g divides f.

Generalizations extend beyond rings. For any poset P, the set Id(P) of all ideals (downward closed and directed subsets) forms a complete lattice under inclusion, connecting order-theoretic and lattice-theoretic perspectives. In lattice theory, the lattice of ideals of a lattice L itself provides a complete lattice and features in dualities and representations.

Applications relate to prime ideals and the spectrum in algebraic geometry, where radical and nilpotent ideals correspond to geometric properties, and to the study of substructures, quotients, and dimension theory. Note: the term ideallattice is occasionally used informally to refer to this lattice of ideals or as a proper name for related software or research projects.