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Note on Primary Ideal Decompositions

Published online by Cambridge University Press:  20 November 2018

P. J. McCarthy
P. J. McCarthy*
Affiliation:
University of Kansas
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Let R be a ring with a unity element. An ideal Q of R is called (right) primary if for ideals A and B of R, AB ⊂ Q and A ⊄ Q imply that Bn ⊂ Q for some positive integer n. If R satisfies the ascending chain condition for ideals (ACC), then R is said to have a Noetherian ideal theory if every ideal of R is an intersection of a finite number of primary ideals. If R is a commutative ring that satisfies the ACC, then R has a Noetherian ideal theory. However, it is known that in general R may satisfy the ACC without having a Noetherian ideal theory (an example of such a ring is given in (2)). Thus there is some interest in conditions that imply that a ring R satisfying the ACC will have a Noetherian ideal theory.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 950 - 952
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Barnes, W. E. and Cunnea, W. M., Ideal decompositions in Noetherian rings, Can. J. Math., 17 (1965), 178–184.Google Scholar
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3. Dilworth, R. P., Non-commutative residuated lattices, Trans. Amer. Math. Soc., 46 (1939), 426–444.Google Scholar
4. Murdoch, D. C., Contributions to noncommutative ideal theory, Can. J. Math., 4 (1952), 43–57.Google Scholar
5. Riley, J. A., Axiomatic primary and tertiary decomposition theory, Trans. Amer. Math. Soc., 105 (1962), 177–201.Google Scholar