Some results on pure ideals and trace ideals of projective modules (Q2141412)

Let \(R\) be a commutative ring with unit. The author investigates \textit{pure ideals} and \textit{trace ideals} of projective \(R\)-modules. An ideal \(I\) of \(R\) is called a pure ideal if the canonical ring morphism \(R \to R/I\) is a flat ring morphism. Theorem 2.1 gives a new characterization of pure ideals in terms of coprime-ness of the ideal \(I\) with the annihilator of its elements. It is shown in Corollary 2.4, that the characterization of the pure ideals can be extended further. In the remaining part of the section, the author studies the notions of \textit{strongly pure} and \textit{quasi-pure} ideals. In Proposition 2.5 the author shows that every strongly pure ideal is a regular ideal. Lemma 2.6 and Proposition 2.7 give sufficient conditions for the rings being zero-dimensional and absolutely flat respectively, in terms of its maximal ideals being quasi-pure and pure ideals respectively. The trace ideal of an \(R\)-module \(M\) is the ideal \(\mathtt{tr}_M(R)=\sum_{f\in \Hom(M,R)}f(M)\). In Theorem 3.1, it is proven that the trace ideal of a projective \(R\)-module \(P\) is generated by the coordinates of the elements of \(P\). As corollaries, the author gives alternative proofs of some well-known results. Corollary 3.3 shows that a submodule of a module \(M\), which is generated by an element \(x\in M\), is projective if and only if \(\mathtt{Ann}(x)\) is generated by an idempotent element.