Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Principal elements of lattices of ideals


Author: P. J. McCarthy
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] K. P. Bogart, Structure theorems for regular local Noether lattices, Michigan Math. J. 15 (1968), 167-176. MR 37 #2642. MR 0227057 (37:2642)
  • [2] -, Distributive local Noether lattices, Michigan Math. J. 16 (1969), 215-223. MR 40 #5513. MR 0252292 (40:5513)
  • [3] R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math. 12 (1962), 481-498. MR 26 #1333. MR 0143781 (26:1333)
  • [4] S. Greco, Sugli ideali frazionari invertibili, Rend. Sem. Mat. Univ. Padova 36 (1966), 315-333. MR 34 #1345. MR 0201461 (34:1345)
  • [5] M. Griffin, Prüfer rings with zero divisors, J. Reine Angew. Math. 239/240 (1969), 55-67. MR 41 #1969. MR 0255527 (41:188)
  • [6] A. Helms, Ein Beitrag zur algebraischen Geometrie, Math. Ann. 111 (1935), 438-458. MR 1513006
  • [7] M. F. Janowitz, Principal multiplicative lattices, Pacific J. Math. 33 (1970), 653-656. MR 0263796 (41:8396)
  • [8] E. W. Johnson, A-transforms and Hilbert functions in local lattices, Trans. Amer. Math. Soc. 137 (1969), 125-139. MR 38 #5675. MR 0237387 (38:5675)
  • [9] P. J. McCarthy, Arithmetical rings and multiplicative lattices, Ann. Mat. Pura Appl. (4) 82 (1969), 267-274. MR 40 #1378. MR 0248124 (40:1378)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13.10

Retrieve articles in all journals with MSC: 13.10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0279080-4
Keywords: Principal element, invertible ideal
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society