Primal decomposition in rings, modules and lattice modules (Q2577760)

Let \(R\) be a commutative ring with \(1\). A proper ideal \(A\)\ of \(R\) is primal if the set of zerodivisors \(Z_{R}(R/A)\) of \(R/A\) is an ideal. Then \(P\) is called the adjoint of \(A\) and \(A\)\ is said to be \(P\)-primal. Note that primary ideals are primal but not conversely. In analogy to primary decompositions, we have primal decompositions (where the intersection is allowed to be infinite). This concept was introduced by \textit{L. Fuchs} [Proc. Am. Math. Soc. 1, 1--6 (1950; Zbl 0041.16501)], who showed that every proper ideal is an intersection of primal ideals. This was further developed by \textit{L. Fuchs, W. Heinzer}, and \textit{B. Olderding} [Trans. Am. Math. Soc. 357, 2771--2798 (2005; Zbl 1066.13003) and Lect. Notes Pure Appl. Math. 236, 189--203 (2004; Zbl 1096.13028)], who showed each ideal has a ``canonical decomposition'' and considered uniqueness questions of this decomposition. \textit{L. Fuchs} and \textit{R. Reis} [Algebra Univers. 50, 341--357 (2003; Zbl 1081.06016)] obtained a similar decomposition of elements of a compactly generated multiplicative lattice. The paper under review points out that the decomposition given by Fuchs and Reis does not reduce to the canonical decomposition given by Fuchs, Heinzer, and Olberding in the case where the multiplicative lattice is the lattice of ideals of a commutative ring and gives two variations of the multiplicative lattice decomposition of Fuchs and Reis. One of these variations does reduce to the decomposition of Fuchs, Heinzer, and Olberding while the other is superior in some ways.