Abstract
In this paper, we introduce a special kind of reduction S, called a symmetric reduction, of a non-basic ideal I in a local ring (R, M). (By definition, S = (b
1 − u
1, i
b
i
,…, b
i−1 − u
i−1, i
b
i
, b
i+1 − u
i+1, i
b
i
,…, b
g
− u
g, i
b
i
)R for i = 1,…, g, where b
1,…, b
g
is an arbitrary basis of I and the u
i, j
are certain units in R such that for i ≠ j in {1,…, g}; note that S depends on the basis, but it has the nice property that I = (b
i
, S)R for i = 1,…, g.) Our main results show that if I is a non-basic ideal in R and if R has an infinite residue field, then: (1) I has proper such reductions S; (2) there exists a positive integer s(I) such that: (a) for each minimal basis of I, I has a symmetric reduction S depending on this basis such that S
I
s(I) = I
s(I)+1; and, (b) if H is any proper reduction of I and H
I
n
= I
n+1, then n ≥ s(I) (so, in particular, s(I) ≤ r(I), the reduction number of I); and, (3) r(I) ≤ s(S
0) + … + s(S
k−1) (where I = S
0 ⊃ S
1 ⊃ … ⊃ S
k
is a chain of ideals such that S
i
is a symmetric reduction of S
i−1 for i = 1,…, k and S
k
is a minimal reduction of I), and the equality holds for certain ideals I and certain such chains of symmetric reductions of I.
AMS (MOS) Subject Classification Numbers:
Acknowledgments
Notes
#Communicated by C. Maxson.