Star operations in extensions of integral domains

Anderson, David F.; El Baghdadi, Said; Zafrullah, Muhammad

doi:10.5802/acirm.39

Anderson, David F.¹ ; El Baghdadi, Said ² ; Zafrullah, Muhammad 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, Tome 2 (2010) no. 2, pp. 87-89.

Résumé

An extension $D \subseteq R$ 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 $D \subseteq R$ is strongly $t$ -compatible (resp., $t$ -compatible) for every overring $R$ of $D$ .

Publié le : 2011-10-20

Zbl

DOI : 10.5802/acirm.39

Classification : 13B02, 13A15, 13G05
Mots clés : Star operation, $t$-linked, $t$-compatible, strongly $t$-compatible, domain extensions, Prüfer domain.

Affiliations des auteurs :

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

@article{ACIRM_2010__2_2_87_0,
     author = {Anderson, David F. and El Baghdadi, Said and Zafrullah, Muhammad},
     title = {Star operations in extensions of integral domains},
     journal = {Actes des rencontres du CIRM},
     pages = {87--89},
     publisher = {CIRM},
     volume = {2},
     number = {2},
     year = {2010},
     doi = {10.5802/acirm.39},
     zbl = {06938587},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/acirm.39/}
}

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
SP  - 87
EP  - 89
VL  - 2
IS  - 2
PB  - CIRM
UR  - http://www.numdam.org/articles/10.5802/acirm.39/
DO  - 10.5802/acirm.39
LA  - en
ID  - ACIRM_2010__2_2_87_0
ER  -

%0 Journal Article
%A Anderson, David F.
%A El Baghdadi, Said
%A Zafrullah, Muhammad
%T Star operations in extensions of integral domains
%J Actes des rencontres du CIRM
%D 2010
%P 87-89
%V 2
%N 2
%I CIRM
%U http://www.numdam.org/articles/10.5802/acirm.39/
%R 10.5802/acirm.39
%G en
%F ACIRM_2010__2_2_87_0

Anderson, David F.; El Baghdadi, Said; Zafrullah, Muhammad. Star operations in extensions of integral domains. Actes des rencontres du CIRM, Tome 2 (2010) no. 2, pp. 87-89. doi : 10.5802/acirm.39. http://www.numdam.org/articles/10.5802/acirm.39/

Bibliographie
Cité par

[1] D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16(1988) 2535–2553. | DOI | MR | Zbl

[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.

[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. | DOI | MR | Zbl

[4] D. F. Anderson, A general theory of class group, Comm. Algebra 16(1988) 805–847. | DOI | MR | Zbl

[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. | Zbl

[6] D. F. Anderson and A. Rykaert, The class group of $D + M$ , J. Pure Appl. Algebra 52(1988) 199–212. | DOI

[7] J. Arnold and J. Brewer, On flat overrings, ideal transforms and generalized transforms of a commutative ring, J. Algebra 18(1971) 254–263. | DOI | MR | Zbl

[8] V. Barucci, S. Gabelli and M. Roitman, The class group of a strongly Mori domain, Comm. Algebra 22(1994) 173–211. | DOI | MR | Zbl

[9] E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form $D + M$ , Michigan Math. J. 20(1973) 79–95. | DOI | MR | Zbl

[10] J. Brewer and E. Rutter, $D + M$ constructions with general overrings, Michigan Math. J. 23(1976) 33–42. | DOI | MR | Zbl

[11] P.-J. Cahen, S. Gabelli and E. G. Houston, Mori domains of integer-valued polynomials, J. Pure Appl. Algebra 153(2000) 1–15. | DOI | MR | Zbl

[12] D. L. Costa, J. L. Mott and M. Zafrullah, The construction $D + X D_{S} [X]$ , J. Algebra 53(1978) 423–439. | DOI | MR | Zbl

[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. | Zbl

[14] E. Davis, Overrings of commutative rings, II: Integrally closed overrings, Trans. Amer. Math. Soc. 110 (1964) 196–212. | DOI | MR | Zbl

[15] D. Dobbs, E. Houston, T. Lucas, and M. Zafrullah, $t$ -linked overrings and Prüfer $v$ -multilpication domains, Comm. Algebra 17(1989) 2835–2852. | DOI | Zbl

[16] D. Dobbs, E. Houston, T. Lucas, M. Roitman and M. Zafrullah, On $t$ -linked overrings, Comm. Algebra 20(1992) 1463–1488. | DOI | MR | Zbl

[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. | DOI

[18] M. Fontana and S. Gabelli, On the class group and the local class group of a pullback, J. Algebra 181(1996) 803–835. | DOI | MR | Zbl

[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. | DOI | Zbl

[20] M. Fontana, J. Huckaba and I. Papick, Prüfer domains, Monographs and Textbooks in Pure and Applied Mathematics 203, Marcel Dekker, New York, 1997.

[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. | Zbl

[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. | Zbl

[23] R. W. Gilmer, Multiplicative Ideal Theory, Marcel-Dekker, New York, 1972. | Zbl

[24] R. Gilmer and W. Heinzer, Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7(1967) 133–150. | DOI | MR | Zbl

[25] R. Gilmer and J. Ohm, Integral domains with quotient overrings, Math. Ann. 153(1964) 813–818. | DOI | MR | Zbl

[26] J. R. Hedstrom and E. G. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18(1980) 37–44. | DOI | MR | Zbl

[27] W. Heinzer, Quotient overrings of integral domains, Mathematika 17(1970) 139–148. | DOI | MR | Zbl

[28] W. Heinzer and I. J. Papick, The radical trace property, J. Algebra 112(1988) 110–121. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[31] J. Huckaba and I. J. Papick, When the dual of an ideal is a ring, Manuscripta Math. 37(1982) 67–85. | DOI | MR | Zbl

[32] B. G. Kang, Prufer v-multiplication domains and the ring $R {[X]}_{N_{v}}$ , J. Algebra 123(1989) 151–170. | DOI | Zbl

[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. | DOI | Zbl

[34] A. Mimouni, Integral domains in which each ideal is a $w$ -ideal, Comm. Algebra 33(2005) 1345–1355. | DOI | Zbl

[35] T. Nishimura, On the $v$ -ideal of an integral domain, Bull. Kyoto Gakugei Univ. (ser. B) 17(1961) 47–50.

[36] J. Querre, Sur les anneaux reflexifs, Can. J. Math. 6(1975) 1222–1228. | DOI | MR | Zbl

[37] J. Querre, Idéaux divisoriels d’un anneau de polynômes, J. Algebra 60(1980) 270–284. | DOI | Zbl

[38] F. Richman, Generalized quotient rings, Proc. Amer. Math. Soc. 16(1965) 794–799. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[41] M. Zafrullah, Well behaved prime t-ideals, J. Pure Appl. Algebra 65(1990) 199–207. | DOI | MR | Zbl

[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. | DOI | Zbl

Cité par Sources :