When every finitely generated regular ideal is finitely presented (Q6548283)

The authors introduce the notion of a regular coherent ring (reg-coherent for short) as a commutative ring such that every finitely generated regular ideal is finitely presented. They investigate the stability of this property under localization and homomorphic image, and its transfer to various contexts of constructions such as trivial ring extensions, pullbacks and amalgamation. Among others, they prove that if \(A\) is an integral domain which is not a field, \(K := qf(A)\), \(E\) is a \(K\)-vector space and \(R := A\ltimes E\) is the trivial ring extension of \(A\) by \(E\), then \(R\) is a reg-coherent ring if and only if \(A\) is coherent. If \((A, M)\) is a local ring and \(E\) is an \(A/M\)-vector space, then \(R\) is reg-coherent. \N\NFor pullback constructions, they prove that if \(T = K + M\) is a local ring, where \(K\) is a field, \(M\) is the maximal ideal of \(T\) satisfying that for each \(m\in M\), there exists \(n\in M\) such that \(mn = 0\) and \(D\) is a subring of \(K\), then \(R = D + M\) is reg-coherent if and only if \(D\) is coherent. Finally, for the amalgamation, they prove that if \(A\) and \(B\) are commutative rings, \(J\) is an ideal of \(B\), \(f: A\longrightarrow B\) is a ring homomorphism and \(R := A\bowtie^{f} J\) is the amalgamation of \(A\) with \(B\) along \(J\) with respect to \(f\), then: \((1)\) If \(f(Reg(A))\subseteq Reg(B)\) and \(A\) is reg-coherent, then \(R\) is reg-coherent.\N\N\((2)\) If \(A\) is a total ring and \(J\subseteq J(B)\) with \(J\subseteq f(A)\) or \(J \) torsion \(A\)-module, then \(R\) is reg-coherent.