chapter  11
15 Pages

Rings with Prime Nilradical

ByAyman Badawi, Thomas G. Lucas

We assume throughout that all rings are commutative with 1 = 0. For such a ring R, we let Z(R) denote the set of zero divisors of R and Nil(R) denote the nilradical. We say that Nil(R) is divided if it compares with each principal ideal of R (see [9] and [2]). If Nil(R) is both divided and a prime ideal, we say that R is a φ-ring. For convenience we let H denote the class of all φ-rings. In [2], [3], [4], [5], and [6] the first-named author investigated this class of rings. In [2] and [4] he introduced the concepts of φ-pseudo-valuation rings and φ-chained rings. Also, D.F. Anderson and the first-named author made further investigation on the class H in [1] and introduced the concepts of φ-Pru¨fer rings and φ-Be´zout rings. See Section 4 for definitions of these specific types of φ-rings. The “φ” in the name refers to the canonical map φ : T (R) → RNil(R) from T (R), the total quotient ring of R, to R localized at Nil(R) which maps a fraction a/b ∈ T (R) to its image in RNil(R).