Polynomial Rings Over Weak Armendariz Rings need not be Weak Armendariz

Abstract

It is proved that there exists a weak Armendriz ring R over which the polynomial ring R[x] is not weak Armendariz. This answers an open question of Liu and Zhao in 2006 in the negative, and eliminates the misunderstanding of the question having a positive solution by Hashemi in 2008.

Key Words:

• Polynomial rings
• Weak Armendariz rings

2010 Mathematics Subject Classification:

• 16D25
• 16N40

ACKNOWLEDGMENTS

The author would like to express his gratitude to Professor A. Smoktunowicz for her helpful suggestions. This work is supported by the Natural Foundation of Shandong Province in China (Y2008A04, ZR2010AM003, and BS2010SF017).

Notes

Communicated by A. Smoktunowicz.