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A multiplicative property of R-sequences and H1-sets

Published online by Cambridge University Press:  24 October 2008

C. P. L. Rhodes
Affiliation:
University College, Cardiff

Extract

Let R be a commutative ring which may not contain a multiplicative identity. A set of elements a1,…,ak in R will be called an H1-set (this notation is explained in section 1) if for each relation r1a1 + … +rkak = 0 (riR) there exist elements sijR such that

where Xl,…,Xk are indeterminates. Any R-sequence is an H1-set, but there do exist H1-sets which are not R-sequences (see section 1). Throughout this note we consider an H1-set a1,…,ak which we suppose to be partitioned into two non-empty sets bl…, br and cl,…, cs. Our main purpose is to show that the ideals B = Rb1 + … + Rbr and C = Rc1 + … + Rcs satisfy BmCn = BmCn for all positive integers m and n (Corollary 1). This generalizes Lemma 2 of Caruth(2) where the result is proved when a1,…, ak is a permutable R-sequence. Our proof involves more detail than is necessary just for this, and we obtain various other properties of H1-sets. In particular we extend the main results of Corsini(3) concerning the symmetric and Rees algebras of a power of the ideal Ra1 +… + Rak (Corollary 3).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

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