Geodesic
Note on Primary Ideal Decompositions
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 950-952

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
McCarthy, P. J. Note on Primary Ideal Decompositions. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 950-952. doi: 10.4153/CJM-1966-094-4
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