Regularity in terms of reductions in local Noether lattices
Author:
Michael E. Detlefsen
Journal:
Proc. Amer. Math. Soc. 43 (1974), 1-7
MSC:
Primary 13A15
DOI:
https://doi.org/10.1090/S0002-9939-1974-0327728-0
MathSciNet review:
0327728
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Abstract | References | Similar Articles | Additional Information
Abstract: An -sequence in a local Noether lattice
is a sequence of words which satisfy certain factorization properties. If
satisfies the union condition, there exist
-sequences which can be extended to minimal bases for the powers of
. Consequently, if
satisfies the union condition,
is regular if and only if
is a minimal reduction.
- [1] Garrett Birkhoff, Lattice theory, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
- [2] Kenneth P. Bogart, Structure theorems for regular local Noether lattices, Michigan Math. J. 15 (1968), 167–176. MR 227057
- [3] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. II, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1967. MR 0218472
- [4] R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math. 12 (1962), 481–498. MR 143781
- [5] E. W. Johnson, 𝐴-transforms and Hilbert functions in local lattices, Trans. Amer. Math. Soc. 137 (1969), 125–139. MR 237387, https://doi.org/10.1090/S0002-9947-1969-0237387-6
- [6]
-,
-transforms in Noether lattices, Dissertation, University of California, Riverside, Calif., 1966.
- [7] D. G. Northcott, Ideal theory, Cambridge Tracts in Mathematics and Mathematical Physics, No. 42, Cambridge, at the University Press, 1953. MR 0058575
- [8] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145–158. MR 59889, https://doi.org/10.1017/s0305004100029194
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1974-0327728-0
Keywords:
Regular local Noether lattice,
free monoid,
reduction,
union condition,
analytically independent,
-sequence
Article copyright:
© Copyright 1974
American Mathematical Society