Principal elements of lattices of ideals
HTML articles powered by AMS MathViewer
- by P. J. McCarthy PDF
- Proc. Amer. Math. Soc. 30 (1971), 43-45 Request permission
Abstract:
The notion of principal element of a commutative multiplicative lattice was introduced by Dilworth. In this note the principal elements of the lattice of ideals of a commutative ring with unity R are characterized as those ideals of R which are finitely generated and locally principal ideals. It follows that a regular ideal of R is a principal element of the lattice of ideals of R if and only if it is invertible.References
- Kenneth P. Bogart, Structure theorems for regular local Noether lattices, Michigan Math. J. 15 (1968), 167–176. MR 227057
- Kenneth P. Bogart, Distributive local Noether lattices, Michigan Math. J. 16 (1969), 215–223. MR 252292
- R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math. 12 (1962), 481–498. MR 143781
- S. Greco, Sugli ideali frazionari invertibili, Rend. Sem. Mat. Univ. Padova 36 (1966), 315–333 (Italian). MR 201461
- Malcolm Griffin, Prüfer rings with zero divisors, J. Reine Angew. Math. 239(240) (1969), 55–67. MR 255527, DOI 10.1515/crll.1969.239-240.55
- Alfred Helms, Ein Beitrag zur algebraischen Geometrie, Math. Ann. 111 (1935), no. 1, 438–458 (German). MR 1513006, DOI 10.1007/BF01472231
- M. F. Janowitz, Principal multiplicative lattices, Pacific J. Math. 33 (1970), 653–656. MR 263796
- E. W. Johnson, $A$-transforms and Hilbert functions in local lattices, Trans. Amer. Math. Soc. 137 (1969), 125–139. MR 237387, DOI 10.1090/S0002-9947-1969-0237387-6
- P. J. McCarthy, Arithmetical rings and multiplicative lattices, Ann. Mat. Pura Appl. (4) 82 (1969), 267–274. MR 248124, DOI 10.1007/BF02410800
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 43-45
- MSC: Primary 13.10
- DOI: https://doi.org/10.1090/S0002-9939-1971-0279080-4
- MathSciNet review: 0279080