Topologically defined classes of going-down domains

Abstract

Let <italic>R</italic> be an integral domain. Our purpose is to study GD (going-down) domains which arise topologically; that is, we investigate how certain going-down assumptions on <italic>R</italic> and its overrings relate to the topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Spec left-parenthesis upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Spec</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {Spec}}(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Many classes of GD domains are introduced topologically, and a systematic study of their behavior under homomorphic images, localization and globalization, integral change of rings, and the “<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D plus upper M"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">D + M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> construction” is undertaken. Also studied, is the algebraic and topological relationships between these newly defined classes of GD domains.