Arkiv för Matematik

  • Ark. Mat.
  • Volume 45, Number 2 (2007), 241-252.

The amalgamated duplication of a ring along a multiplicative-canonical ideal

Marco D’Anna and Marco Fontana

Full-text: Open access

Abstract

After recalling briefly the main properties of the amalgamated duplication of a ring R along an ideal I, denoted by $R\bowtie I$, see M. D’Anna and M. Fontana, to appear in J. Algebra Appl., we restrict our attention to the study of the properties of $R\bowtie I$, when I is a multiplicative canonical ideal of R, see W. J. Heinzer, J. A. Huckaba and I. J. Papick, Comm. Algebra. In particular, we study when every regular fractional ideal of $R\bowtie I$ is divisorial.

Article information

Source
Ark. Mat., Volume 45, Number 2 (2007), 241-252.

Dates
Received: 8 June 2006
Revised: 29 September 2006
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898983

Digital Object Identifier
doi:10.1007/s11512-006-0038-1

Mathematical Reviews number (MathSciNet)
MR2342602

Zentralblatt MATH identifier
1143.13002

Rights
2007 © Institut Mittag-Leffler

Citation

D’Anna, Marco; Fontana, Marco. The amalgamated duplication of a ring along a multiplicative-canonical ideal. Ark. Mat. 45 (2007), no. 2, 241--252. doi:10.1007/s11512-006-0038-1. https://projecteuclid.org/euclid.afm/1485898983


Export citation

References

  • Bazzoni, S., Divisorial domains, Forum Math. 12 (2000), 397–419.
  • Bazzoni, S. and Salce, L., Warfield domains, J. Algebra 185 (1996), 836–868.
  • Bruns, W. and Herzog, J., Cohen–Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993.
  • D’Anna, M., A construction of Gorenstein rings, J. Algebra 306 (2006), 507–519.
  • D’Anna, M. and Fontana, M., An amalgamated duplication of a ring along an ideal: the basic properties, to appear in J. Algebra Appl.
  • Fossum, R., Commutative extensions by canonical modules are Gorenstein rings, Proc. Amer. Math. Soc. 40 (1973), 395–400.
  • Fossum, R. M., Griffith, P. A. and Reiten, I., Trivial Extensions of Abelian Categories, Springer, Berlin–Heidelberg, 1975.
  • Fuchs, L. and Salce, L., Modules over Non-Noetherian Domains, Math. Surveys Monogr. 84, Amer. Math. Soc., Providence, RI, 2001.
  • Heinzer, W. J., Huckaba, J. A. and Papick, I. J., m-canonical ideals in integral domains, Comm. Algebra 26 (1998), 3021–3043.
  • Huckaba, J. A., Commutative Rings with Zero Divisors, Monogr. Textbooks Pure Appl. Math. 117, Marcel Dekker, New York, 1988.
  • Kunz, E., Beispiel: Die kanonische Idealklasse eines eindimensionalen Cohen–Macaulay-Rings, in Der kanonische Modul eines Cohen–Macaulay-Rings (Sem. Lokale Kohomologietheorie von Grothendieck, Univ. Regensburg, Regensburg, 1970/1971), Lecture Notes in Math. 238, pp. 17–24, 103, Springer, Berlin–Heidelberg, 1971.
  • Maimani, H. R. and Yassemi, S., Zero-divisor graphs of amalgamated duplication of a ring along an ideal, Preprint, 2006.
  • Matlis, E., 1-dimensional Cohen–Macaulay Rings, Springer, Berlin–Heidelberg, 1973.
  • Nagata, M., Local Rings, Interscience Tracts in Pure and Applied Mathematics 13, Interscience-Wiley, New York–London, 1962.
  • Olberding, B., Stability, duality, 2-generated ideals and a canonical decomposition of modules, Rend. Sem. Mat. Univ. Padova 106 (2001), 261–290.
  • Reiten, I., The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417–420.