A generalization of (D + M)-construction

Let [Formula: see text] be a local domain with [Formula: see text] such that [Formula: see text]. Each subring [Formula: see text] of [Formula: see text] determines a subring [Formula: see text] of [Formula: see text]. This construction arises frequently in Algebra, essentially in connection with counter-examples. In this paper, we use the results of the authors [D. F. Anderson and A. Badawi, On [Formula: see text]-Prüfer rings and [Formula: see text]-Bezout rings, Houston J. Math. 30(2) (2004) 331–343; A. Badawi, On [Formula: see text]-chained rings and [Formula: see text]-pseudo-valuation rings, Houston J. Math. 27(4) (2001) 725–736; A. Badawi, On nonnil-Noetherian rings, Comm. Algebra 31(4) (2003) 1669–1677; A. Badawi, On pseudo-almost valuation domains, Comm. Algebra 35(4) (2007) 1167–1181; E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form [Formula: see text], Michigan Math. J. 20 (1973) 79–95; J. W. Brewer and E. A. Rutter, [Formula: see text] constructions with general overrings, Michigan Math. J. 23(1) (1976) 33–42; A. Esmaeelnezhad and P. Sahandi, On [Formula: see text]-pseudo almost valuation rings, Bull. Korean Math. Soc. 52(3) (2015) 935–946; R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers on Pure and Applied Mathematics, Vol. 90 (Queen’s University, Kingston, 1992); B. Khoualdia and A. Benhissi, Nonnil-coherent rings, Beitr. Algebra Geom. 57(2) (2016) 297–305; G. H. Tang, F. G. Wang and W. Zhao, On [Formula: see text]-Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] to generalize the above construction in the context of rings with prime Nilradical.