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 semilocal $ {\rm OP}$-rings


Author: Yukitoshi Hinohara
Journal: Proc. Amer. Math. Soc. 32 (1972), 16-20
MSC: Primary 13.95
DOI: https://doi.org/10.1090/S0002-9939-1972-0289502-1
MathSciNet review: 0289502
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The notion of OP-rings was introduced by D. Lissner. A commutative ring R is called an OP-ring if, for any $ n \geqq 2$, any vector of $ {R^n}$ is an outer product of $ n - 1$ vectors of $ {R^n}$. Recently J. Towber proved that any local ring is an OP-ring if and only if the maximal ideal is generated by two elements. The main result in the present paper is a generalization to semilocal rings of the above theorem proved by Towber for local rings. The author's argument does not rely on Towber's theorem however, and so provides a new and very elementary proof of that result.


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

  • [1] N. Bourbaki, Algèbre commutative, Chaps. 1, 2, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR 36 #146.
  • [2] O. Forster, Über die Anzahl der Erzeugenden eines Ideals in einem Noetherschen Ring, Math. Z. 84 (1964), 80-87. MR 29 #1231. MR 0163932 (29:1231)
  • [3] D. Lissner, Matrices over polynomial rings, Trans. Amer. Math. Soc. 98 (1961), 285-305. MR 23 #A171. MR 0122839 (23:A171)
  • [4] -, Outer product rings, Trans. Amer. Math. Soc. 116 (1965), 526-535. MR 32 #4145. MR 0186687 (32:4145)
  • [5] -, OP rings and Seshadri's theorem, J. Algebra 5 (1967), 362-366. MR 34 #4295. MR 0204453 (34:4295)
  • [6] D. Lissner and A. Geramita, Towber rings, J. Algebra 15 (1970), 13-40. MR 41 #1713. MR 0257059 (41:1713)
  • [7] J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Séminaire P. Dubreil, M.-L. Dubriel-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23, Secrétariat Mathématique, Paris, 1958. MR 31 #1277.
  • [8] R. G. Swan, The number of generators of a module, Math. Z. 102 (1967), 318-322. MR 36 #1434. MR 0218347 (36:1434)
  • [9] J. Towber, Complete reducibility in exterior algebras over free modules, J. Algebra 10 (1968), 299-309. MR 38 #1116. MR 0232793 (38:1116)
  • [10] -, Local rings with the outer product property, Illinois J. Math. 14 (1970), 194-197. MR 41 #193. MR 0255532 (41:193)
  • [11] O. Zariski and P. Samuel, Commutative algebra. Vol. 1, University Series in Higher Math., Van Nostrand, Princeton, N.J., 1958. MR 19, 833. MR 0090581 (19:833e)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13.95

Retrieve articles in all journals with MSC: 13.95


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0289502-1
Keywords: OP-ring, unimodular vector, outer product, semilocal ring, local ring, Jacobson radical, principal ideal, principal ideal ring, special principal ideal ring, projective module
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society