Some countability conditions on commutative ring extensions
Some countability conditions on commutative ring extensions
If<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"><mml:semantics><mml:mi>S</mml:mi><mml:annotation encoding="application/x-tex">S</mml:annotation></mml:semantics></mml:math></inline-formula>is a finitely generated unitary extension ring of the commutative ring<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"><mml:semantics><mml:mi>R</mml:mi><mml:annotation encoding="application/x-tex">R</mml:annotation></mml:semantics></mml:math></inline-formula>, then<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"><mml:semantics><mml:mi>S</mml:mi><mml:annotation encoding="application/x-tex">S</mml:annotation></mml:semantics></mml:math></inline-formula>cannot be expressed as the union of a strictly ascending sequence<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper R Subscript n Baseline right-brace Subscript n equals 1 …