Pullbacks of $\mathcal{C}$-hereditary domains
Bull. Korean Math. Soc. 2018 Vol. 55, No. 4, 1093-1101
Published online March 8, 2018
Printed July 31, 2018
Yongyan Pu, Gaohua Tang, Fanggui Wang
Panzhihua University, Guangxi Teacher's Education University, Sichuan Normal University
Abstract : Let $(RDTF,M)$ be a Milnor square. In this paper, it is proved that $R$ is a $\mathcal{C}$-hereditary domain if and only if both $D$ and $T$ are $\mathcal{C}$-hereditary domains; $R$ is an almost perfect domain if and only if $D$ is a field and $T$ is an almost perfect domain; $R$ is a Matlis domain if and only if $T$ is a Matlis domain. Furthermore, to give a negative answer to Lee$^,$s question, we construct a counter example which is a $\mathcal{C}$-hereditary domain $R$ with $w.gl.\dim(R)=\infty$.
Keywords : $\mathcal{C}$-hereditary domain, Matlis domain, almost perfect domain, Milnor square
MSC numbers : 13C99, 13A15
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