Further results on reticulated rings

Abstract

For more details about the reticulation LR of a ring, see Simmons [9]. For any ideal I of the ring R, let D(I) be the ideal generated by {D(a) : a E I ) in LR. For any ideal J of the lattice LR, let D-l (J ) = {a E R: D(a) E J). Trivially, D-I (J ) is an ideal of R. The reticulation of a ring was investigated by Simmons [9] in order to show that a lo t of ring theoretic properties have analogues in lattice theory and vice versa. In this paper we continue this theme and answer a couple of questions raised by Simmons [9]. Then we proceed to prove further results in that direction. Let Id(R) be the lattice of all ideals of the ring R. It should be noted that this lattice is not necessarily distributive. Let RId(R) be the distributive lattice of radical ideals of the ring R. For a lattice L, let Id(L) be the lattice of ideals of L. From now on, LR will denote the reticulation of R. We start by quoting a theorem that was given by Johnstone [7], p. 194.