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

Full-text PDF

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**0227057****[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**0143781****[5]**E. W. Johnson,*𝐴-transforms and Hilbert functions in local lattices*, Trans. Amer. Math. Soc.**137**(1969), 125–139. MR**0237387**, 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**0059889**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
13A15

Retrieve articles in all journals with MSC: 13A15

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