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1976•Canadian Journal of MathematicsOpen access

Localization in Non-Commutative Noetherian Rings

Bruno J. Müller

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Abstract

To construct a well behaved localization of a noetherian ringRat a semiprime ideal S, it seems necessary to assume that the set (S)of moduloSregular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ringRsin addition: one calls such 5classical.To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal.

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To construct a well behaved localization of a noetherian ringRat a semiprime ideal S, it seems necessary to assume that the set (S)of moduloSregular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ringRsin addition: one calls such 5classical.To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal.

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Available abstract

To construct a well behaved localization of a noetherian ringRat a semiprime ideal S, it seems necessary to assume that the set (S)of moduloSregular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ringRsin addition: one calls such 5classical.To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal.

Key concepts: Mathematics, Semiprime, Semiprime ring, Radical of a ring, Ideal (ethics), Associated prime, Pure mathematics, Minimal ideal

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