Weakly Prime Elements in Multiplicative Lattices

Abstract

In this article, we study weakly prime elements and almost prime elements in multiplicative lattices. Next we characterize weak π-lattices and weak principal element lattices. We also establish new characterizations for π-domains and principal element lattices in terms of almost prime elements. Finally, we show that in a Noether lattice L, every proper element of L is a finite product of almost prime elements if and only if L is a finite direct product of principal element domains, special principal element lattices, and special product of almost prime elements lattices.

Key Words:

• Almost prime element
• Invertible element
• π-Lattice
• Principal element
• Principal element lattice
• Regular lattice
• Weak π-lattice
• Weakly prime element
• Weak principal element lattice

2010 Mathematics Subject Classification:

• Primary 06F10
• Secondary 06F05, 13A15

Notes

Communicated by V. Gould.