In this paper, we study a rings in which every maximal ideal is finitely generated provided some of its power is finitely generated. This notion is raised by Gilmer in 1971 and Roitman shows that coherent domains satisfy this property in 2001. We establish the transfer of this notion to direct products, trivial ring extensions, pullbacks, and the amalgamation of rings. Our results generate new families of examples of non-coherent rings (with zerodivisors) satisfy this condition.