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Zbl 0668.13005
Cahen, Paul-Jean
Couples d'anneaux partageant un idÃ©al. (Couples of rings sharing an ideal).
(French)
[J] Arch. Math. 51, No.6, 505-514 (1988). ISSN 0003-889X; ISSN 1420-8938/e

Let A, B be commutative rings with identity. The author calls the pair (A,B) a proper couple if A is a proper subring of B and A and B have a nonzero ideal I in common. In this case, A and B are said to share the ideal I. For example, if B is a valuation ring of the form $K+M$, where M is the maximal ideal of B and K is the residue field, and $A=D+M$ where D is a proper subring of K then (A,B) is a proper couple, sharing the ideal M. This D$+M$ construction has given a rich source of counterexamples in commutative ring theory. \par In section 1 of the paper, stability properties of the construction are generalized to proper couples (A,B). These include conditions under which A is Noetherian, integrally closed or local. - In section $2,$ the height of the shared ideal I in A is compared with its height in B and a formula for dim(A), the Krull dimension of A, in terms of dim(B) is determined in the case where every prime ideal of B containing I is maximal. - $Section\quad 3$ investigates questions of height and dimension in the extension of the proper couple (A,B), sharing the ideal I, to the proper couple (A[X],B[X]), sharing the ideal I[X], where X is any finite set of indeterminates. - The final section looks at the lifting of prime ideals of A to B in the proper couple (A,B) and the construction of non-catenary rings. The paper is well-provided with examples.
[J.Clark]
MSC 2000:
*13B02 Extension theory (commutative rings)
13A15 Ideals; multiplicative ideal theory
13A18 Valuations and their generalizations

Keywords: proper couple; valuation ring; $D+M$ construction; shared ideal; Krull dimension; lifting of prime ideals

Cited in: Zbl 1094.13501

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