Star operations in extensions of integral domains

David F. Anderson[1]; Said El Baghdadi[2]; Muhammad Zafrullah[3]

  • [1] Department of Mathematics, University of Tennessee Knoxville, TN 37996, USA
  • [2] Department of Mathematics, Faculté des Sciences et Techniques P.O. Box 523, Beni Mellal, Morocco
  • [3] 57 Colgate Street, Pocatello, ID 83201, USA

Actes des rencontres du CIRM (2010)

  • Volume: 2, Issue: 2, page 87-89
  • ISSN: 2105-0597

Abstract

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An extension D ⊆ R of integral domains is strongly t -compatible (resp., t -compatible) if ( I R ) - 1 = ( I - 1 R ) v (resp., ( I R ) v = ( I v R ) v ) for every nonzero finitely generated fractional ideal I of D . We show that strongly t -compatible implies t -compatible and give examples to show that the converse does not hold. We also indicate situations where strong t -compatibility and its variants show up naturally. In addition, we study integral domains D such that D ⊆ R is strongly t -compatible (resp., t -compatible) for every overring R of D .

How to cite

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Anderson, David F., El Baghdadi, Said, and Zafrullah, Muhammad. "Star operations in extensions of integral domains." Actes des rencontres du CIRM 2.2 (2010): 87-89. <http://eudml.org/doc/196282>.

@article{Anderson2010,
abstract = {An extension $D \subseteq R$ of integral domains is strongly$t$-compatible (resp., $t$-compatible) if $(IR)^\{-1\} = (I^\{-1\}R)_\{v\}$ (resp., $(IR)_\{v\} = (I_\{v\}R)_\{v\})$ for every nonzero finitely generated fractional ideal $I$ of $D$. We show that strongly $t$-compatible implies $t$-compatible and give examples to show that the converse does not hold. We also indicate situations where strong $t$-compatibility and its variants show up naturally. In addition, we study integral domains $D$ such that $D \subseteq R$ is strongly $t$-compatible (resp., $t$-compatible) for every overring $R$ of $D$.},
affiliation = {Department of Mathematics, University of Tennessee Knoxville, TN 37996, USA; Department of Mathematics, Faculté des Sciences et Techniques P.O. Box 523, Beni Mellal, Morocco; 57 Colgate Street, Pocatello, ID 83201, USA},
author = {Anderson, David F., El Baghdadi, Said, Zafrullah, Muhammad},
journal = {Actes des rencontres du CIRM},
keywords = {Star operation; $t$-linked; $t$-compatible; strongly $t$-compatible; domain extensions; Prüfer domain},
language = {eng},
number = {2},
pages = {87-89},
publisher = {CIRM},
title = {Star operations in extensions of integral domains},
url = {http://eudml.org/doc/196282},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Anderson, David F.
AU - El Baghdadi, Said
AU - Zafrullah, Muhammad
TI - Star operations in extensions of integral domains
JO - Actes des rencontres du CIRM
PY - 2010
PB - CIRM
VL - 2
IS - 2
SP - 87
EP - 89
AB - An extension $D \subseteq R$ of integral domains is strongly$t$-compatible (resp., $t$-compatible) if $(IR)^{-1} = (I^{-1}R)_{v}$ (resp., $(IR)_{v} = (I_{v}R)_{v})$ for every nonzero finitely generated fractional ideal $I$ of $D$. We show that strongly $t$-compatible implies $t$-compatible and give examples to show that the converse does not hold. We also indicate situations where strong $t$-compatibility and its variants show up naturally. In addition, we study integral domains $D$ such that $D \subseteq R$ is strongly $t$-compatible (resp., $t$-compatible) for every overring $R$ of $D$.
LA - eng
KW - Star operation; $t$-linked; $t$-compatible; strongly $t$-compatible; domain extensions; Prüfer domain
UR - http://eudml.org/doc/196282
ER -

References

top
  1. D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16(1988) 2535–2553. Zbl0672.13001MR955324
  2. D. D. Anderson, D.F. Anderson and M. Zafrullah, Rings between D[X] and K[X], Houston J. Math. 17(1)(1991) 109–129. Zbl0736.13015MR1107192
  3. D. D. Anderson, E. Houston and M. Zafrullah, t-linked extensions, the t -class group and Nagata’s theorem, J. Pure Appl. Algebra 86(1993) 109–124. Zbl0777.13002MR1215640
  4. D. F. Anderson, A general theory of class group, Comm. Algebra 16(1988) 805–847. Zbl0648.13002MR932636
  5. D. F. Anderson, S. El Baghdadi and S. Kabbaj, The class group of A + X B [ X ] domains, in: Advances in commutative ring theory, Lecture Notes in Pure and Appl. Math. 205, Dekker, New York, 1999, pp. 73–85. Zbl0979.13010MR1767451
  6. D. F. Anderson and A. Rykaert, The class group of D + M , J. Pure Appl. Algebra 52(1988) 199–212. Zbl0668.13013MR952078
  7. J. Arnold and J. Brewer, On flat overrings, ideal transforms and generalized transforms of a commutative ring, J. Algebra 18(1971) 254–263. Zbl0218.13019MR276215
  8. V. Barucci, S. Gabelli and M. Roitman, The class group of a strongly Mori domain, Comm. Algebra 22(1994) 173–211. Zbl0828.13012MR1255678
  9. E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M , Michigan Math. J. 20(1973) 79–95. Zbl0239.13001MR323782
  10. J. Brewer and E. Rutter, D + M constructions with general overrings, Michigan Math. J. 23(1976) 33–42. Zbl0318.13007MR401744
  11. P.-J. Cahen, S. Gabelli and E. G. Houston, Mori domains of integer-valued polynomials, J. Pure Appl. Algebra 153(2000) 1–15. Zbl0978.13010MR1781539
  12. D. L. Costa, J. L. Mott and M. Zafrullah, The construction D + X D S [ X ] , J. Algebra 53(1978) 423–439. Zbl0407.13003MR506224
  13. D. L. Costa, J. L. Mott and M. Zafrullah, Overrings and dimensions of general D + M constructions, J. Natur. Sci. and Math. 26(2) (1986) 7–14. Zbl0621.13010MR871405
  14. E. Davis, Overrings of commutative rings, II: Integrally closed overrings, Trans. Amer. Math. Soc. 110 (1964) 196–212. Zbl0128.26005MR156868
  15. D. Dobbs, E. Houston, T. Lucas, and M. Zafrullah, t -linked overrings and Prüfer v -multilpication domains, Comm. Algebra 17(1989) 2835–2852. Zbl0691.13015MR1025612
  16. D. Dobbs, E. Houston, T. Lucas, M. Roitman and M. Zafrullah, On t -linked overrings, Comm. Algebra 20(1992) 1463–1488. Zbl0791.13005MR1157918
  17. S. El Baghdadi, On TV-domains, in: Commutative Algebra and its Applications, de Gruyter Proceedings in Mathematics, de Gruyter, Berlin, 2009, pp. 207–212. Zbl1177.13008MR2606286
  18. M. Fontana and S. Gabelli, On the class group and the local class group of a pullback, J. Algebra 181(1996) 803–835. Zbl0871.13006MR1386580
  19. M. Fontana and J. Huckaba, Localizing systems and semistar operations, in: Non-Noetherian Commutative Ring Theory (S. Chapman and S. Glaz, Eds.) Math. Appl. 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 169–197. Zbl1047.13002MR1858162
  20. M. Fontana, J. Huckaba and I. Papick, Prüfer domains, Monographs and Textbooks in Pure and Applied Mathematics 203, Marcel Dekker, New York, 1997. Zbl0861.13006MR1413297
  21. M. Fontana and N. Popescu, On a class of domains having Prüfer integral closure: the ℱ QR-domains, in: Commutative Ring Theory, Lecture Notes in Pure and Appl. Math. 185, Dekker, New York, 1997, pp. 303–312. Zbl0879.13011MR1422488
  22. S. Gabelli, On Nagata’s theorem for the class group II, in: Commutative algebra and algebraic geometry, Lecture Notes in Pure and Appl. Math. 206, Dekker, New York, 1999, pp. 117–142. Zbl0945.13005MR1702102
  23. R. W. Gilmer, Multiplicative Ideal Theory, Marcel-Dekker, New York, 1972. Zbl0248.13001MR427289
  24. R. Gilmer and W. Heinzer, Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7(1967) 133–150. Zbl0166.30601MR223349
  25. R. Gilmer and J. Ohm, Integral domains with quotient overrings, Math. Ann. 153(1964) 813–818. Zbl0128.26004MR159835
  26. J. R. Hedstrom and E. G. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18(1980) 37–44. Zbl0462.13003MR578564
  27. W. Heinzer, Quotient overrings of integral domains, Mathematika 17(1970) 139–148. Zbl0201.37202MR265334
  28. W. Heinzer and I. J. Papick, The radical trace property, J. Algebra 112(1988) 110–121. Zbl0641.13001MR921967
  29. E. Houston, personal communication to M. Zafrullah. 
  30. E. Houston and M. Zafrullah, Integral domains in which each t -ideal is divisorial, Michigan Math. J. 35(1988) 291–300. Zbl0675.13001MR959276
  31. J. Huckaba and I. J. Papick, When the dual of an ideal is a ring, Manuscripta Math. 37(1982) 67–85. Zbl0484.13001MR649566
  32. B. G. Kang, Prufer v-multiplication domains and the ring R [ X ] N v , J. Algebra 123(1989) 151–170. Zbl0668.13002MR1000481
  33. T. G. Lucas, Examples built with D + M, A + XB[X], and other pullback constructions, in: Non-Noetherian Commutative Ring Theory (S. Chapman and S. Glaz, Eds.) Math. Appl. 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 341–368. Zbl1005.13002MR1858170
  34. A. Mimouni, Integral domains in which each ideal is a w -ideal, Comm. Algebra 33(2005) 1345–1355. Zbl1079.13016MR2149062
  35. T. Nishimura, On the v -ideal of an integral domain, Bull. Kyoto Gakugei Univ. (ser. B) 17(1961) 47–50. Zbl0266.13016MR142574
  36. J. Querre, Sur les anneaux reflexifs, Can. J. Math. 6(1975) 1222–1228. Zbl0335.13010MR414537
  37. J. Querre, Idéaux divisoriels d’un anneau de polynômes, J. Algebra 60(1980) 270–284. Zbl0441.13012MR575795
  38. F. Richman, Generalized quotient rings, Proc. Amer. Math. Soc. 16(1965) 794–799. Zbl0145.27406MR181653
  39. A. Rykaert, Sur le groupe des classes et le groupe local des classes d’un anneau intègre, Ph.D Thesis, Universite Claude Bernard de Lyon I, 1986. 
  40. M. Zafrullah, Finite conductor domains, Manuscripta Math. 24(1978) 191–203. Zbl0383.13013MR485847
  41. M. Zafrullah, Well behaved prime t-ideals, J. Pure Appl. Algebra 65(1990) 199–207. Zbl0705.13001MR1068255
  42. M. Zafrullah, Putting t-invertibility to use, in: Non-Noetherian Commutative Ring Theory (S. Chapman and S. Glaz, Eds.) Math. Appl. 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 429–457. Zbl0988.13003MR1858174

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