Reductions of Ideals in Pullbacks

This paper deals with reductions of ideals in various settings of pullback constructions. Precisely, we investigate reductions of several types of ideals in both generic and classical pullbacks. We also characterize pullbacks where reductions of a class of ideals extend to reductions of their respective extended ideals. All results are illustrated with original examples.

[1]  A. Mimouni,et al.  On t-reductions of ideals in pullbacks , 2016, 1607.06705.

[2]  S. Kabbaj,et al.  t-Reductions and t-integral closure of ideals , 2016, 1602.07041.

[3]  A. Mimouni,et al.  Core of ideals in integral domains , 2016 .

[4]  S. Kabbaj.  On the dimension theory of polynomial rings over pullbacks , 2006, math/0606686.

[5]  Craig Huneke,et al.  Integral closure of ideals, rings, and modules , 2006 .

[6]  A. Mimouni,et al.  Trace Properties and Pullbacks , 2003 .

[7]  W. Vasconcelos,et al.  On the integral closure of ideals , 1998, math/0210229.

[8]  Stefania Gabelli,et al.  Coherentlike conditions in pullbacks. , 1997 .

[9]  M. Fontana,et al.  On the Class Group and the Local Class Group of a Pullback , 1996 .

[10]  A. Bouvier,et al.  On Jaffard domains , 1988 .

[11]  M. Fontana.  Topologically defined classes of commutative rings , 1980 .

[12]  David F. Anderson,et al.  Pairs of Rings with the Same Prime Ideals , 1980, Canadian Journal of Mathematics.

[13]  E. Houston,et al.  Pseudo-valuation domains. , 1978 .

[14]  J. Brewer,et al.  $D+M$ constructions with general overrings. , 1976 .

[15]  J. H. Hays.  Reductions of ideals in Prüfer domains , 1975 .

[16]  Eduardo Bastida,et al.  Overrings and divisorial ideals of rings of the form $D+M$. , 1973 .

[17]  J. H. Hays.  Reductions of ideals in commutative rings , 1973 .

[18]  D. G. Northcott,et al.  Reductions of ideals in local rings , 1954, Mathematical Proceedings of the Cambridge Philosophical Society.