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Radicals Of Polynomial Rings

Published online by Cambridge University Press:  20 November 2018

S. A. Amitsur
S. A. Amitsur*
Affiliation:
Hebrew University, Jerusalem, Israel
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Introduction. Let R be a ring and let R[x] be the ring of all polynomials in a commutative indeterminate x over R. Let J(R) denote the Jacobson radical (5) of the ring R and let L(R) be the lower radical (4) of R. The main object of the present note is to determine the radicals J(R[x]) and L(R[x]). The Jacobson radical J(R[x]) is shown to be a polynomial ring N[x] over a nil ideal N of R and the lower radical L(R[x]) is the polynomial ring L(R)[x].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 8 , 1956 , pp. 355 - 361
Copyright
Copyright © Canadian Mathematical Society 1956

References

1. Amitsur, S. A., A general theory of radicals II, Amer. J. Math., 76 (1954), 100–125.Google Scholar
2. Amitsur, S. A., A general theory of radicals III, Amer. J. Math., 76 (1954), 126–136.Google Scholar
3. Amitsur, S. A., Algebras over infinite fields, Proc. Amer. Math. Soc. (to be published).Google Scholar
4. Baer, R., Radical ideals, Amer. J. Math., 69 (1947), 46–58.Google Scholar
5. Jacobson, N., The radical and semi simplicity for arbitrary rings, Amer. J. Math., 67 (1945), 300–320.Google Scholar
6. Levitzki, J., A theorem on polynomial identities, Proc. Amer. Math. Soc, 1 (1950), 331–333.Google Scholar
7. Snapper, E., Completely primary rings I, Ann. Math., 52 (1950), 666–693.Google Scholar