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Content Algebras

Published online by Cambridge University Press:  20 November 2018

David E. Rush
David E. Rush*
Affiliation:
Department of Mathematics, University of California, Riverside, California 92521
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Let R be a commutative ring with identity and let X be an indeterminate. In [5] and [16] it was shown that if f, g∈R[X], then for some integer n≥l, c(f)n+1c(g) = c(g)nc(fg), where c(h) denotes the additive subgroup of JR generated by the coefficients of h ∈ R[X]. Actually the statements in [5J and [16] are not so general as this; however, the proofs are. Specifically, in [16] Mertens considers only the case that R is a polynomial ring over the integers, but this gives the result for any ring by specializing the coefficients of f and g. In [5] Dedekind considers only the case that JR is a ring of algebraic integers, but his proof is completely general. Further, the above formula then holds if one lets c(h) denote the S-submodule of R generated by the coefficients of h ∈ R[X], S a subring of R, and it is in this form that it usually appears, especially the case S = R. Dedekind′s very elegant proof is reproduced in [15, p. 9, Lemma 6.1].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 3 , 01 September 1978 , pp. 329 - 334
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Arnold, J. T., Gilmer, R., Heinzer, W. J., Some countability conditions in a commutative ring, Illinois Journal Math. 21 (1977), 648-665.Google Scholar
2. Arnold, J. T. and Gilmer, R., On the contents of polynomials, Proc. Amer. Math. Soc. 24 (1970), 556-562.Google Scholar
3. Bass, H., Connell, E. H. and Wright, D. L., Locally polynomial algebras are symmetric, Bull. Amer. Math. Soc. 82 (1976), 719-720.Google Scholar
4. Bourbaki, N., Commutative Algebra, Addison-Wesley, Reading, Mass., 1972.Google Scholar
5. Dedekind, R., Über einen arithemetischen satz von Gauss, Mitt. Deutsch Math. Ges. Prague, 1892, 1-11; Gesammelte Werke XXII, Vol. 2. Google Scholar
6. Eakin, P. M. and Heinzer, W. J., A cancellation problem for rings, Conf. on Commutative Algebra (Lawrence, Kansas, 1972), Lecture Notes in Math., vol. 311, Springer-Verlag, Berlin and New York, 1973, pp. 61-77.Google Scholar
7. Eakin, P. and Silver, J., Rings which are almost polynomial rings, Trans. Amer. Math. Soc. 174 (1972), 425-449.Google Scholar
8. Ferrand, D., Les Modules projectifs de type fini sur un anneau de polynômes sur un corps sont libres (d'après Quillen et Suslin), Séminaire Bourbaki, 28 (1975), (484-01)-(484-20).Google Scholar
9. Gilmer, R., Multiplicative Ideal Theory, M. Dekker, New York, 1972.Google Scholar
10. Gilmer, R., Some applications of the Hilfssatz von Dedekind-Mertens, Math. Scand. 20 (1967), 240-244.Google Scholar
11. Gilmer, R., Polynomial rings over a commutative Von Neumann regular ring, Proc. Amer. Math. Soc. 49 (1975), 294-296.Google Scholar
12. Gilmer, R., Grams, A., and Parker, T., Zero divisors in power series rings, J. Reine Angew. Math. 278/279 (1975), 145-146.Google Scholar
13. Gilmer, R. and Parker, T., Divisibility properties in semigroup rings, Mich. Math. J. 21 (1974), 65-86.Google Scholar
14. Krull, W., Idealtheorie, Springer-Verlag, Berlin-Heidelberg-New York, 1935.Google Scholar
15. Lefschetz, S., Algebraic Geometry, Princeton University Press, Princeton, New Jersey, 1953.Google Scholar
16. Mertens, F., Über einen algebraischen satz, S.-B. Akad. Wiss. Wein Abtheilung IlalOl (1892), 1560-1566.Google Scholar
17. Mott, J. L. and Schexnayder, M., Exact sequences of semi-value groups, J. Reine Angew. Math. 283/284 (1976), 388-401.Google Scholar
18. Nagata, M., Local Rings, Robert E. Krieger Pub. Co., Huntington, New York, 1975.Google Scholar
19. Northcott, D. G., A generalization of a theorem on the contents of polynomials, Proc. Cambridge Philos. Soc. 55 (1959), 282-288.Google Scholar
20. Ohm, J. and Rush, D. E., The finiteness of I when R[X]/I is flat, Trans. Amer. Math. Soc. 171 (1972), 377-408.Google Scholar
21. Ohm, J. and Rush, D. E., Content modules and algebras, Math. Scand. 31 (1972), 49-68.Google Scholar
22. Prüfer, H., Untersuchungen über Teilbarkeitseigenschaften in Körpem, J. Reine Angew. Math. 168 (1932), 1-36.Google Scholar
23. Quillen, D., Projective modules over polynomial rings, Inv. Math. 36 (1976), 167-171.Google Scholar
24. Rush, D. E., Remarks on flat modules, Mich. Math. J. 23 (1976), 193-201.Google Scholar
25. Tang, H. T., Gauss’ lemma, Proc. Amer. Math. Soc. 35 (1972), 372-376.Google Scholar
26. Vasconcelos, W. V., Simple flat extensions II, Math. Z. 129 (1972), 157-161.Google Scholar
27. Wright, D. L., Algebras which resemble symmetric algebras, Queen's Papers in Pure and Appl. Math., No. 42, Kingston, Ontario, Canada (1975), pp. 226-240.Google Scholar