2020/10/07 by Azarang, Alborz
#13A15 13G05 13F05 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2010.03248
Let R be a commutative ring, we say that A⊆ Spec(R) has prime avoidance property, if I⊆ \bigcupP\inAP for an ideal I of R, then there exists P\inA such that I⊆ P. We exactly determine when A⊆ Spec(R) has prime avoidance property. In particular, if A has prime avoidance property, then A is compact. For certain classical rings we show the converse holds (such as Bezout rings, QR-domains, zero-dimensional rings and C(X)). We give an example of a compact set A⊆ Spec(R), where R is a Prufer domain, which has not P.A-property. Finally, we show that if V,V1,…, Vn are valuation domains for a field K and V[x]\nsubseteq \bigcupi=1n Vi for some x∈ K, then there exists v∈ V such that v+x∉ \bigcupi=1n Vi.