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.
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.