Principal multiplicative lattices
Melvin F. Janowitz · Pacific Journal of Mathematics · 1970
McCarthy has recently proved that if R is a Noetherian ring with unity, then every ideal of R is a principal element of L(R), the lattice of ideals of R 9 if and only if R is a multiplication ring.It is shown here that an arbitrary commutative ring R with unity is a Noetherian multiplication ring if and only if every ideal of R is a principal element of L(R).1* M-lattices* The basic terminology and notation will follow that of [2] and [3].It will be assumed throughout the paper that L denotes a complete commutative and residuated multiplicative lattice.The results of this section, though well known, have apparently not been previously published.They are included here because they are needed to prove the results of § 2.