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Commutative rings whose quotients are Goldie

Published online by Cambridge University Press:  18 May 2009

Victor P. Camillo
Victor P. Camillo
Affiliation:
University of Iowa, Iowa City, Iowa 52240, U.S.A.
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All rings considered here have units. A (non-commutative) ring is right Goldieif it has no infinite direct sums of right ideals and has the ascending chain condition on annihilator right ideals. A right ideal A is an annihilator if it is of the form {a ∈ R/xa = 0 for all x ∈ X}, where X is some subset of R. Naturally, any noetherian ring is Goldie, but so is any commutative domain, so that the converse is not true. On the other hand, since any quotient ring of a noetherian ring is noetherian, it is true that every quotient is Goldie. A reasonable question therefore is the following: must a ring, such that every quotient ring is Goldie, be noetherian? We prove the following theorem:

Theorem. A commutative ring is noetherian if and only if every quotient is Goldie.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 16 , Issue 1 , March 1975 , pp. 32 - 33
Copyright
Copyright © Glasgow Mathematical Journal Trust 1975

References

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