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)

 
 

 

On the imbedding of a direct product into a zero-dimensional commutative ring


Authors: Robert Gilmer and William Heinzer
Journal: Proc. Amer. Math. Soc. 106 (1989), 631-636
MSC: Primary 13B99
DOI: https://doi.org/10.1090/S0002-9939-1989-0969521-2
MathSciNet review: 969521
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper addresses questions related to results of M. Arapovic concerning imbeddability of a commutative unitary ring $ R$ in a zero-dimensional ring. The case where $ R$ is a product of zero-dimensional rings is of special interest. We show (1) if the zero ideal of $ R$ admits a unique representation as an irredundant intersection of (strongly primary) ideals, then $ R$ need not be imbeddable in a zero-dimensional ring, and (2) for a family $ \left\{ {{R_\alpha }} \right\}$ of zero-dimensional rings, $ R = \prod {R_\alpha }$ is imbeddable in a zero-dimensional ring if and only if $ R$ itself is zero-dimensional.


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

  • [1] M. Arapovic, Characterizations of the 0-dimensional rings, Glas. Mat. 18 (1983), 39-46. MR 710383 (84m:13005a)
  • [2] -, The minimal 0-dimensional overrings of commutative rings, Glas. Mat. 18 (1983), 47-52. MR 710384 (84m:13005b)
  • [3] -, On the imbedding of a commutative ring into a 0-dimensional ring, Glas. Mat. 18 (1983), 53-59. MR 710385 (84m:13005c)
  • [4] R. Gilmer and W. Heinzer, Products of commutative rings and zero-dimensionality, in preparation.
  • [5] J. Huckaba, On valuation rings that contain zero divisors, Proc. Amer. Math. Soc. 40 (1973), 9-19. MR 0318134 (47:6683)
  • [6] -, Commutative rings with zero divisors, Marcel Dekker, New York, 1988. MR 938741 (89e:13001)
  • [7] P. Maroscia, Sur les anneux de dimension zero, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 56 (1974), 451-459. MR 0389877 (52:10707)
  • [8] N. Popescu and C. Vraciu, Sur la structure des anneaux absolutement plats commutatifs, J. Algebra 40 (1976), 364-383. MR 0439834 (55:12716)
  • [9] D. Underwood, On some uniqueness questions in primary representation of ideals, J. Math. Kyoto Univ. 9 (1969), 69-94. MR 0246865 (40:134)
  • [10] O. Zariski and P. Samuel, Commutative algebra, vol. I, Springer, Berlin, 1975. MR 0389876 (52:10706)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13B99

Retrieve articles in all journals with MSC: 13B99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0969521-2
Keywords: Imbedding, zero-dimensional rings, products of commutative rings
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society