Nonnil-coherent rings

Abstract

Let R be a commutative ring with \(1 \ne 0\) and let \(H = \{ R | R\;is\;a\;commutative\;ring\;and\;Nil(R)\;is\;a\;divided\;prime\, {ideal}\;of\;R\}\). If \(R \in H\), then R is called a \(\phi \)-ring. In this paper, we introduce a new class of rings that is closely related to the class of coherent rings. A ring R is called nonnil-coherent if every finitely generated nonnil ideal of R is finitely presented. We show that many of the properties of coherent rings are also true for nonnil-coherent rings. We show that the localization of a nonnil-coherent rings is not necessary a coherent ring. Also, we define the concept of \(\phi \)-coherent rings and we show that the nonnil-coherent rings and the \(\phi \)-coherent rings are not necessary equivalent. Finally, we use the idealization construction to give examples of \(\phi \)-coherent rings that are not coherent rings.