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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Artinian subrings of a commutative ring


Authors: Robert Gilmer and William Heinzer
Journal: Trans. Amer. Math. Soc. 336 (1993), 295-310
MSC: Primary 13E10; Secondary 12D15, 12F99, 13A99
DOI: https://doi.org/10.1090/S0002-9947-1993-1102887-7
MathSciNet review: 1102887
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Abstract: Given a commutative ring $ R$, we investigate the structure of the set of Artinian subrings of $ R$. We also consider the family of zero-dimensional subrings of $ R$. Necessary and sufficient conditions are given in order that every zero-dimensional subring of a ring be Artinian. We also consider closure properties of the set of Artinian subrings of a ring with respect to intersection or finite intersection, and the condition that the set of Artinian subrings of a ring forms a directed family.


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

  • [C$ _{1}$] I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54-106. MR 0016094 (7:509h)
  • [C$ _{2}$] -, Commutative rings with restricted minimum condition, Duke Math. J. 17 (1950), 27-42. MR 0033276 (11:413g)
  • [E] P. Eakin, The converse to a well-known theorem on Noetherian rings, Math. Ann. 177 (1968), 278-282. MR 0225767 (37:1360)
  • [G$ _{1}$] R. Gilmer, Integral domains with Noetherian subrings, Comment. Math. Helv. 45 (1970), 129-134. MR 0262216 (41:6826)
  • [G$ _{2}$] -, Domains with integrally closed subrings, Math Japon. 16 (1971), 9-11. MR 0302641 (46:1785)
  • [G$ _{3}$] -, Multiplicative ideal theory, Marcel-Dekker, New York, 1972. MR 0427289 (55:323)
  • [GH$ _{1}$] R. Gilmer and W. Heinzer, Noetherian pairs and hereditarily Noetherain rings, Arch. Math. 41 (1983), 131-138. MR 719415 (85d:13022)
  • [GH$ _{2}$] -, On the imbedding of a direct product into a zero-dimensional commutative ring, Proc. Amer. Math. Soc. 106 (1989), 631-637. MR 969521 (89m:13005)
  • [GH$ _{3}$] -, Products of commutative rings and zero-dimensionality, Trans. Amer. Math. Soc. 331 (1992), 663-680. MR 1041047 (92h:13015)
  • [GH$ _{4}$] -, Zero-dimensionality in commutative rings, Proc. Amer. Math. Soc. 115 (1992), 881-893. MR 1095223 (92j:13011)
  • [H] J. Huckaba, Commutative rings with zero divisors, Marcel-Dekker, New York, 1988. MR 938741 (89e:13001)
  • [J] N. Jacobson, Lectures in abstract algebra, vol. 3, Van Nostrand, Princeton, N.J., 1964. MR 0172871 (30:3087)
  • [L] J. Lambek, Lectures on rings and modules, Blaisdell, Waltham, Mass., 1966. MR 0206032 (34:5857)
  • [M] P. Maroscia, Sur les anneaux de dimension zero, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 56 (1974), 451-459. MR 0389877 (52:10707)
  • [N] M. Nagata, Local rings, Interscience, New York, 1962. MR 0155856 (27:5790)
  • [W$ _{1}$] A. Wadsworth, Pairs of domains where all intermediate domains are Noetherian, Trans. Amer. Math. Soc. 195 (1974), 201-211. MR 0349665 (50:2158)
  • [W$ _{2}$] -, Hilbert subalgebras of finitely generated algebras, J. Algebra 43 (1976), 298-304. MR 0427295 (55:329)
  • [ZS] O. Zariski and P. Samuel, Commutative algebra, vol. I, Springer, New York, 1975. MR 0389876 (52:10706)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1102887-7
Keywords: Artinian subring, zero-dimensional subring, directed union of Artinian subrings, Noetherian ring, finite spectrum, hereditarily Noetherian ring, zero-dimensional pair, coefficient field, coefficient ring, absolutely algebraic field
Article copyright: © Copyright 1993 American Mathematical Society

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