Finite unions of overrings of an integral domain

Avoidance principle and intersection property for a class of rings

“…Interestingly, our proof is much simpler, slicker and direct. In particular, we prove that [1], Corollary 8 holds for arbitrary domain, that is, the assumption of Bézout domain can be dropped. Our next motive is to discuss the question raised by Gottlieb in [1], Example 7.…”

“…In particular, we prove that [1], Corollary 8 holds for arbitrary domain, that is, the assumption of Bézout domain can be dropped. Our next motive is to discuss the question raised by Gottlieb in [1], Example 7. In [1], Proposition 5, Gottlieb proved that if A is a local overring of an integral domain R such that A ⊇ R p1 ∩ R p2 ∩ .…”

“…All rings considered below are commutative with nonzero identity. Let S denote the saturation of the multiplicative closed subset S of a ring R, let Nil(R) denote the set of all nilpotent elements in R, let Z(R) denote the set of all zero divisors in R, and let T (R) denote the total quotient ring of R. In this note, we extend the work of Gottlieb, see [1] from finite unions to infinite unions of overrings of an integral domain. The main theme of Gottlieb's paper was the following: If A, A 1 , A 2 , .…”

“…Our next motive is to discuss the question raised by Gottlieb in [1], Example 7. In [1], Proposition 5, Gottlieb proved that if A is a local overring of an integral domain R such that A ⊇ R p1 ∩ R p2 ∩ . .…”

“…∩ R pn , then A ⊇ R pi for some i. This motivated him to raise the question whether we can replace R p 's by S −1 R in [1], Proposition 5. In [1], Example 7, he showed that the answer is no.…”