Regular elements in multiplicative lattices
Throughout, \(L\) is a not necessarily modular multiplicative lattice (\(L\) has a commutative associative product that distributes over joins and has greatest element 1 as a multiplicative identity) that is join-generated by a multiplicatively closed set of compact elements with 1 compact and join-generated by a set of principal elements (\(e\) is called principal if \(a\wedge be = ((a : e)\wedge b)e\) and \((ae\vee b): e=(b:e)\vee a)\). An element \(a\) is called invertible if \(a\) is principal and \(0 : a = 0\), and \(a\) is called regular if it contains an invertible element. The paper gives a number of characterizations of the lattices \(L\) in which every regular element is invertible, such as every regular element is a product of prime elements, or every invertible element is contained in only finitely many prime elements, and for every regular maximal prime \(m\) of \(L\), \(m\) is locally principal, and \(\bigwedge_{k=1}^\infty m^k\) is nonregular. Several new characterization of Dedekind lattices (every element not contained in a minimal prime element is weak meet principal (i.e., satisfies the first of the two principal element identities)) and WI-lattices (every compact element \(a\in L\) is principal and \((0 : (0 : a))\vee (0 : a) = 1)\) are given. Applications are given to commutative rings.
scientific article; zbMATH DE number 5500517