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Krull dimension and invertible ideals in Noetherian rings

1976; Cambridge University Press; Volume: 20; Issue: 2 Linguagem: Inglês

10.1017/s0013091500010580

1464-3839

T. H. Lenagan,

Rings, Modules, and Algebras

In this note we consider the question: If R is a right Noetherian ring and I is an invertible ideal of R , how do the Krull dimensions of various modules, factor rings and over-rings of R , connected with I , compare with the Krull dimension of R ? This question is prompted by results in ( 5 ) and ( 6 ). In comparing the Krull dimension of the ring R with that of the ring R / I , the best result would be that the Krull dimension of the ring R is exactly one greater than that of the ring R / I . This result is not true in general; however, we see, in Theorem 2.4, that if the invertible ideal is contained in the Jacobson radical the result holds. In the general case we find it is necessary to introduce an over-ring T of R generated by the inverse I −1 of I . We then see that the Krull dimension of R is the larger of two possibilities: ( a ) Krull dimension of R / I plus one or ( b ) Krull dimension of T . In order to prove this result we construct a strictly increasing map from the poset of right ideals of R to the cartesian product of the poset of right ideals of T with a poset of certain infinite sequences of right ideals of R / I .