The purpose of this paper is to introduce a new class of rings that is closely related to the classes of pseudo valuation rings (PVRs) and pseudo-almost valuation domains (PAVDs). A commutative ring R is said to be

-ring if its nilradical Nil(R) is both prime and comparable with each principal ideal. The name is derived from the natural map

from the total quotient ring T(R) to R localized at Nil(R). A prime ideal P of a

-ring R is said to be a

-pseudo-strongly prime ideal if, whenever

and

, then there exists an integer

such that either

or

. If each prime ideal of R is a

-pseudo strongly prime ideal, then we say that R is a

-pseudo-almost valuation ring (

-PAVR). Among the properties of

-PAVRs, we show that a quasilocal

-ring R with regular maximal ideal M is a

-PAVR if and only if V = (M : M) is a

-almost chained ring with maximal ideal

. We also investigate the overrings of a

-PAVR.