Almost valuation property in bi-amalgamations and pairs of rings

Abstract

This paper examines the transfer of the almost valuation property to various constructions of ring extensions such as bi-amalgamations and pairs of rings. Namely, Sec. 2 studies the transfer of this property to bi-amalgamation rings. Our results cover previous known results on duplications and amalgamations, and provide the construction of various new and original examples satisfying this property. Section 3 investigates pairs of integral domains where all intermediate rings are almost valuation rings. As a consequence of our results, we provide necessary and sufficient conditions for a pair (R, T), where R arises from a (T, M, D) construction, to be an almost valuation pair. Furthermore, we study the class of maximal non-almost valuation subrings of their quotient field.