IRREDUCIBLE IDEALS IN RINGS

Abstract

It is well known that the ideals of any ring form an algebraic lattice under the set inclusion ordering, in which the finitely generated ideals are precisely the compact elements. Strongly irreducible ideals of a ring were studied by W.J. Heinzer, L.J. Ratliff and D.E. Rush [3] and α−irreducible and α−strongly irreducible ideals of a ring were characterized by X. Lu and H. Qin [8].  In this paper, we extend these results for elements of a general algebraic lattice and obtain the results on ideals of rings and on submodules of an R−module as consequences of our general results.  Also, we characterize algebraic lattices satisfying the ascending chain condition.