An extension of integral domains is strongly -compatible (resp., -compatible) if (resp., for every nonzero finitely generated fractional ideal of . We show that strongly -compatible implies -compatible and give examples to show that the converse does not hold. We also indicate situations where strong -compatibility and its variants show up naturally. In addition, we study integral domains such that is strongly -compatible (resp., -compatible) for every overring of .
@article{ACIRM_2010__2_2_87_0, author = {David F. Anderson and Said El Baghdadi and Muhammad Zafrullah}, 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}, language = {en}, url = {https://acirm.centre-mersenne.org/articles/10.5802/acirm.39/} }
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. https://acirm.centre-mersenne.org/articles/10.5802/acirm.39/
[1] D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16(1988) 2535–2553.
[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 -class group and Nagata’s theorem, J. Pure Appl. Algebra 86(1993) 109–124.
[4] D. F. Anderson, A general theory of class group, Comm. Algebra 16(1988) 805–847.
[5] D. F. Anderson, S. El Baghdadi and S. Kabbaj, The class group of domains, in: Advances in commutative ring theory, Lecture Notes in Pure and Appl. Math. 205, Dekker, New York, 1999, pp. 73–85.
[6] D. F. Anderson and A. Rykaert, The class group of , J. Pure Appl. Algebra 52(1988) 199–212.
[7] J. Arnold and J. Brewer, On flat overrings, ideal transforms and generalized transforms of a commutative ring, J. Algebra 18(1971) 254–263.
[8] V. Barucci, S. Gabelli and M. Roitman, The class group of a strongly Mori domain, Comm. Algebra 22(1994) 173–211.
[9] E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form , Michigan Math. J. 20(1973) 79–95.
[10] J. Brewer and E. Rutter, constructions with general overrings, Michigan Math. J. 23(1976) 33–42.
[11] P.-J. Cahen, S. Gabelli and E. G. Houston, Mori domains of integer-valued polynomials, J. Pure Appl. Algebra 153(2000) 1–15.
[12] D. L. Costa, J. L. Mott and M. Zafrullah, The construction , J. Algebra 53(1978) 423–439.
[13] D. L. Costa, J. L. Mott and M. Zafrullah, Overrings and dimensions of general constructions, J. Natur. Sci. and Math. 26(2) (1986) 7–14.
[14] E. Davis, Overrings of commutative rings, II: Integrally closed overrings, Trans. Amer. Math. Soc. 110 (1964) 196–212.
[15] D. Dobbs, E. Houston, T. Lucas, and M. Zafrullah, -linked overrings and Prüfer -multilpication domains, Comm. Algebra 17(1989) 2835–2852.
[16] D. Dobbs, E. Houston, T. Lucas, M. Roitman and M. Zafrullah, On -linked overrings, Comm. Algebra 20(1992) 1463–1488.
[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.
[18] M. Fontana and S. Gabelli, On the class group and the local class group of a pullback, J. Algebra 181(1996) 803–835.
[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.
[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 QR-domains, in: Commutative Ring Theory, Lecture Notes in Pure and Appl. Math. 185, Dekker, New York, 1997, pp. 303–312.
[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.
[23] R. W. Gilmer, Multiplicative Ideal Theory, Marcel-Dekker, New York, 1972.
[24] R. Gilmer and W. Heinzer, Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7(1967) 133–150.
[25] R. Gilmer and J. Ohm, Integral domains with quotient overrings, Math. Ann. 153(1964) 813–818.
[26] J. R. Hedstrom and E. G. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18(1980) 37–44.
[27] W. Heinzer, Quotient overrings of integral domains, Mathematika 17(1970) 139–148.
[28] W. Heinzer and I. J. Papick, The radical trace property, J. Algebra 112(1988) 110–121.
[29] E. Houston, personal communication to M. Zafrullah.
[30] E. Houston and M. Zafrullah, Integral domains in which each -ideal is divisorial, Michigan Math. J. 35(1988) 291–300.
[31] J. Huckaba and I. J. Papick, When the dual of an ideal is a ring, Manuscripta Math. 37(1982) 67–85.
[32] B. G. Kang, Prufer v-multiplication domains and the ring , J. Algebra 123(1989) 151–170.
[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.
[34] A. Mimouni, Integral domains in which each ideal is a -ideal, Comm. Algebra 33(2005) 1345–1355.
[35] T. Nishimura, On the -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.
[37] J. Querre, Idéaux divisoriels d’un anneau de polynômes, J. Algebra 60(1980) 270–284.
[38] F. Richman, Generalized quotient rings, Proc. Amer. Math. Soc. 16(1965) 794–799.
[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.
[41] M. Zafrullah, Well behaved prime t-ideals, J. Pure Appl. Algebra 65(1990) 199–207.
[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.