Geodesic
A Remark on Coherent Overrings
Canadian mathematical bulletin, Tome 21 (1978) no. 3, pp. 373-375

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

Throughout this note, let R be a (commutative integral) domain with quotient field K. A domain S satisfying R ⊆ S ⊆ K is called an overring of R, and by dimension of a ring we mean Krull dimension. Recall [1] that a commutative ring is said to be coherent if each finitely generated ideal is finitely presented.In [2], as a corollary of a more general theorem, Davis showed that if each overring of a domain R is Noetherian, then the dimension of R is at most 1. (This corollary is the converse of a version of the Krull-Akizuki Theorem [5, Theorem 93], and can also be proved directly by using the existence of valuation rings dominating finite chains of prime ideals [4, Corollary 16.6].) It is our purpose to prove that if R is Noetherian and each overring of R is coherent, then the dimension of £ is at most 1. We shall also indicate some related questions and examples. 
Papick, Ira J. A Remark on Coherent Overrings. Canadian mathematical bulletin, Tome 21 (1978) no. 3, pp. 373-375. doi: 10.4153/CMB-1978-067-4
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