From the introduction: All rings considered below are commutative with unit, and all modules are unital. If \(n\) is a nonnegative integer, we say that an \(R\)-module \(M\) is \(n\)-presented if there is an exact sequence \(F_n\to F_{n-1} \to\cdots\to F_0\to M\to 0\) of \(R\)-modules in which each \(F_i\) is finitely generated and free. In particular, “0-presented” means finitely generated and “1-presented” means finitely presented. A ring \(R\) is coherent if and only if each finitely generated ideal of \(R\) is finitely presented; equivalently, if and only if each finitely presented \(R\)-module is 2-presented.
Let \(n\) be a positive integer. We say that \(R\) is \(n\)-coherent (as a ring) if each \((n-1)\)-presented ideal of \(R\) is \(n\)-presented; and that \(R\) is a strong \(n\)-coherent ring if each \(n\)-presented \(R\)-module is \((n+1)\)-presented. Thus, the 1-coherent rings are just the coherent rings. In general, any strong \(n\)-coherent ring is \(n\)-coherent. The converse holds if \(n=1\), but it is an open question for \(n\geq 2\). Notice that each Bezout domain \(R\) is \(n\)-coherent for each \(n>1\); indeed, each \((n-1)\)-presented ideal of \(R\) is principal and hence infinitely-presented (in the obvious sense). Moreover, each Noetherian ring is \(n\)-coherent for any \(n\geq 1\).
Section 2 begins, more generally, by defining \(n\)-coherent modules for each integer \(n\geq 1\). A ring \(R\) is an \(n\)-coherent ring if and only if \(R\) is an \(n\)-coherent \(R\)-module. Several results on transfer of \(n\)-coherence are developed in section 2, and these are used in section 3 to develop examples of \(n\)-coherent rings (and, more generally, to study associated properties) in two pullback contexts.
For the entire collection see [Zbl 0855.00015].