Primal decomposition in rings, modules and lattice modules

Abstract.

In a work by L. Fuchs, W. Heinzer and B. Olberding, a decomposition of ideals in a commutative ring as intersections of primal isolated component ideals is investigated. In subsequent work by L. Fuchs and R. Reis, these ideas are developed in multiplicative lattices. The object of this note is to point out that, when specialized to the lattice of ideals of a commutative ring, the decomposition of L. Fuchs and R. Reis does not give the decomposition obtained in the paper by Fuchs, Heinzer and Olberding, and to give two variations of the decomposition of Fuchs and Reis. One of these variations, when specialized to the lattice of ideals of a ring, does give the decomposition obtained by Fuchs, Heinzer and Olberding, and the other one gives a decomposition which is superior in some ways.