Distributivity and IM-lattices. (English) Zbl 1177.06004

Călugăreanu, Grigore; Khazal, Reyadh R.

Ital. J. Pure Appl. Math. 15, 175-184 (2004).

Summary: Notice that in the diamond (i.e., the 5-element nondistributive lattice) the intersection of the (three) maximal elements is not irredundant, and a lattice is not distributive if it contains a diamond. Hence, connections between the distributivity of a lattice and the irredundancy of the intersection of its family of maximal elements seem plausible.
In this paper, the authors prove that, under natural hypotheses, distributivity is equivalent with certain conditions on maximal elements. Applications to the distributivity of the lattice of all ideals of a semiprimitive ring with identity are given.

MSC:

06B05	Structure theory of lattices
06D05	Structure and representation theory of distributive lattices
16D60	Simple and semisimple modules, primitive rings and ideals in associative algebras

Keywords:

distributivity; maximal elements; lattice; ideals; semiprimitive ring