Nonnil-Noetherian pairs of the form (R, R[X]) and some related results

Abstract

The rings considered in this paper are commutative with identity and are nonzero. Let R be a ring. An ideal I of R is said to be nonnil if it is not contained in the nilradical of R. We say that R is nonnil-Noetherian (resp., nonnil-Laskerian) if each proper nonnil ideal of R is finitely generated (resp., admits a primary decomposition).  Whenever T is an extension ring of R, we assume that R contains the identity element of T.   Let T be an extension ring of R. We say that (R, T) is a Nonnil-Noetherian pair (resp., Nonnil-Laskerian pair) if f A is nonnil-Noetherian (resp., nonnil-Laskerian) for any intermediate ring A between R and T.  This paper aims to characterize R such that (R, R[X]) is a nonnil-Noetherian pair (resp., nonnil-Laskerian pair), where R[X] is the polynomial ring in one variable X over R.  Also, this paper aims to characterize R such that each intermediate ring A between R and R[X] posses a property which is related to being nonnil-Noetherian (resp., nonnil-Laskerian).