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1994•Communications in AlgebraRequires access

A principal ideal theorem analogue for modules over commutative rings

Seleena M. George, Roy McCasland, Patrick F. Smith

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Abstract

The Principal Ideal Theorem states that if Re is a commutative Noetherian ring and ? is a prime ideal of Re which is minimal over a principal ideal then Pe has height at most 1. Also, if Re is a (not necessarily Noetherian) UFD and Pe is a prime ideal of Re minimal over a principal ideal then Pe has height at most 1. We shall show that there are analogues for modules over commutative rings, but they hold only in special cases.

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What this paper is about

The Principal Ideal Theorem states that if Re is a commutative Noetherian ring and ? is a prime ideal of Re which is minimal over a principal ideal then Pe has height at most 1. Also, if Re is a (not necessarily Noetherian) UFD and Pe is a prime ideal of Re minimal over a principal ideal then Pe has height at most 1. We shall show that there are analogues for modules over commutative rings, but they hold only in special cases.

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Available abstract

The Principal Ideal Theorem states that if Re is a commutative Noetherian ring and ? is a prime ideal of Re which is minimal over a principal ideal then Pe has height at most 1. Also, if Re is a (not necessarily Noetherian) UFD and Pe is a prime ideal of Re minimal over a principal ideal then Pe has height at most 1. We shall show that there are analogues for modules over commutative rings, but they hold only in special cases.

Key concepts: Mathematics, Ideal (ethics), Primary ideal, Principal ideal, Prime ideal, Principal (computer security), Noetherian, Radical of an ideal

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