Journal article
Commutative rings whose quotients are Goldie
Glasgow mathematical journal, Vol.16(1), pp.32-33
03/1975
DOI: 10.1017/S0017089500002470
Abstract
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.
Details
- Title: Subtitle
- Commutative rings whose quotients are Goldie
- Creators
- Victor P Camillo - University of Iowa
- Resource Type
- Journal article
- Publication Details
- Glasgow mathematical journal, Vol.16(1), pp.32-33
- DOI
- 10.1017/S0017089500002470
- ISSN
- 0017-0895
- eISSN
- 1469-509X
- Publisher
- Cambridge University Press
- Number of pages
- 2
- Language
- English
- Date published
- 03/1975
- Academic Unit
- Mathematics
- Record Identifier
- 9984240767102771
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