All rings considered in this paper are commutative, associative, and have an identity. If A and B are ideals in a ring, then B is a reduction of A if B C A and if BAtn = An+l for some positive integer n. An ideal is basic if it has no reductions. These definitions were considered in local rings by Northcott and Rees; this paper considers them in more general rings. Basic ideals in Noetherian rings are characterized to the extent that they are characterized in local rings. It is shown that elements of the principal class generate a basic ideal in a Noetherian ring. Prufer domains do not have the basic ideal property, that is, there may exist ideals which are not basic; however, a characterization of PruYfer domains can be given in terms of basic ideals. A domain is Prufer if and only if every finitely generated ideal is basic.
[1]
R. Gilmer,et al.
On the Number of Generators of an Invertible Ideal
,
1970
.
[2]
E. Davis.
Further remarks on ideals of the principal class
,
1968
.
[3]
R. Gilmer,et al.
Multiplicative ideal theory
,
1968
.
[4]
E. Davis.
Ideals of the principal class, R-sequences and a certain monoidal transformation
,
1967
.
[5]
R. Gilmer,et al.
Primary Ideals and Prime Power Ideals
,
1966,
Canadian Journal of Mathematics.
[6]
D. Rees,et al.
A note on reductions of ideals with an application to the generalized Hilbert function
,
1954,
Mathematical Proceedings of the Cambridge Philosophical Society.
[7]
D. G. Northcott,et al.
Reductions of ideals in local rings
,
1954,
Mathematical Proceedings of the Cambridge Philosophical Society.