Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-20T06:30:04.183Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiplication Ideals, Multiplication Rings, and the Ring R(X)

Published online by Cambridge University Press:  20 November 2018

D. D. Anderson
D. D. Anderson*
Affiliation:
University of Missouri, Columbia, Missouri
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R be a commutative ring with an identity. An ideal A of R is called a multiplication ideal if for every ideal B ⊆ A there exists an ideal C such that B = AC. A ring R is called a multiplication ring if all its ideals are multiplication ideals. A ring R is called an almost multiplication ring if RM is a multiplication ring for every maximal ideal M of R. Multiplication rings and almost multiplication rings have been extensively studied—for example, see [4; 8; 9; 11; 12; 15; and 16].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 4 , 01 August 1976 , pp. 760 - 768
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Anderson, D. D., A remark on the lattice of ideals of a Priifer domain, Pacific J. Math. 57 (1975), 323–324.Google Scholar
2. Arnold, J. T., On the ideal theory of the Kronecker function ring and the domain D(X), Can. J. Math. 21 (1969), 558–563.Google Scholar
3. Brewer, J. V. and Rutter, E. A., A note on finitely generated ideals which are locally principal, Proc. Amer. Math. Soc. 31 (1972), 429–432.Google Scholar
4. Butts, H. S. and Phillips, R. C., Almost multiplication rings, Can. J. Math. 17 (1965), 267–277.Google Scholar
5. Camillo, V. P., Semiher editory polynomial rings, Proc. Amer. Math. Soc. If.5 (1974), 173–174.Google Scholar
6. Estes, D. and Ohm, J., Stable range in commutative rings, J. Algebra 7 (1967), 343–362.Google Scholar
7. Gilmer, R. W., Multiplicative ideal theory (Marcel Dekker, New York, 1972).Google Scholar
8. Gilmer, R. W. and Mott, J. L., Multiplication rings as rings in which ideals with prime radical are primary, Trans. Amer. Math. Soc. 114 (1965), 40–52.Google Scholar
9. Griffin, M., Multiplication rings via their total quotient rings, Can. J. Math. 26 (1974), 430–449.Google Scholar
10. Hinkle, G. W. and Huckaba, J. A., The generalized Kronecker function ring and the ring R(X), preprint.Google Scholar
11. Janowitz, M. F., Principal multiplicative lattices, Pacific J. Math. 33 (1970), 653–656.Google Scholar
12. Larsen, M. D. and McCarthy, P. J., Multiplicative theory of ideals (Academic Press, New York, 1971).Google Scholar
13. McCarthy, P. J., Principal elements of lattices of ideals, Proc. Amer. Math. Soc. 30 (1971), 43–45.Google Scholar
14. McCarthy, P. J., The ring of polynomials over a von Neumann regular ring, Proc. Amer. Math. Soc. 39 (1973), 253–254.Google Scholar
15. Mott, J. L., Equivalent conditions for a ring to be a multiplication ring, Can. J. Math. 16 (1964), 429–434.Google Scholar
16. Mott, J. L., Multiplication rings containing only finitely many minimal primes, J. Sci. Hiroshima Univ. Ser. A-I, 33 (1969), 73–83.Google Scholar
17. Smith, W. W., Projective ideals of finite type, Can. J. Math. 21 (1969), 1057–1061.Google Scholar
18. Gilmer, R. and Parker, T., Semigroup rings as Prilfer rings, Duke Math. J. J+l (1974), 219–230.Google Scholar