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Dedekind's Invention of Ideals

Dedekind's Invention of Ideals HAROLD M. EDWARDS The present-day formulation of ideal theory is quite standardized. A subset of a commutative ring with unit is called an ideal if it is closed under addition and under multiplication by ring elements. The product of two ideals of a ring is the ideal consisting of all elements of the ring that can be written as sums of products a • b in which a is in the first ideal and b is in the second. The ring itself is an ideal and every ideal A of a ring R has the trivial factorization A = RA. An ideal A is called prime if it has no nontrivial factorizations, that is, if A = BC implies B = R or C = R. Under suitable conditions on R, every ideal can be written as a product of prime ideals and this representation is unique, up to the order of the factors. This is almost exactly the way that Dedekind himself formulated the theory. He was dealing specifically with the ring of integers of an algebraic number field and did not use the term "ring", which was introduced later by Hilbert, and instead of speaking of a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Dedekind's Invention of Ideals

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/15.1.8
Publisher site
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Abstract

HAROLD M. EDWARDS The present-day formulation of ideal theory is quite standardized. A subset of a commutative ring with unit is called an ideal if it is closed under addition and under multiplication by ring elements. The product of two ideals of a ring is the ideal consisting of all elements of the ring that can be written as sums of products a • b in which a is in the first ideal and b is in the second. The ring itself is an ideal and every ideal A of a ring R has the trivial factorization A = RA. An ideal A is called prime if it has no nontrivial factorizations, that is, if A = BC implies B = R or C = R. Under suitable conditions on R, every ideal can be written as a product of prime ideals and this representation is unique, up to the order of the factors. This is almost exactly the way that Dedekind himself formulated the theory. He was dealing specifically with the ring of integers of an algebraic number field and did not use the term "ring", which was introduced later by Hilbert, and instead of speaking of a

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1983

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