ON 2-ABSORBING PRIMARY AND WEAKLY 2-ABSORBING ELEMENTS IN MULTIPLICATIVE LATTICES

Abstract

In this paper, we introduce the concept of 2-absorbing primary and weakly 2-absorbing primary elements which are generalizations of primary and weakly primary elements in multiplicative lattices.  Let L be a multiplicative lattice.  A proper element q of L is said to be a (weakly) 2-absorbing primary element of L if whenever a, b, c ∈ L with (0 ≠ abc ≤ q) abc ≤ q implies either ab ≤ q or ac ≤ \(\sqrt{q}\) or bc ≤ \(\sqrt{q}\).  Some properties of 2-absorbing primary and weakly 2-absorbing primary elements are presented and relations among prime, primary, 2-absorbing, weakly 2-absorbing, 2-absorbing primary and weakly 2-absorbing primary elements are investigated.  Furthermore, we determine 2-absorbing primary elements in some special lattices and give a new characterization for principal element domains in terms of 2-absorbing primary elements.