Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47886220200701On the Arens regularity of a module action and its extensions19319810981610.22034/kjm.2020.109816ENSedigheBarootkoobDepartment of Mathematics, University of Bojnord, P.O. Box 1339, Bojnord, Iran.Journal Article20190626It is known that if the second dual $A^{**}$ of a Banach algebra $A$ is Arens regular, then $A$ is Arens regular itself. However, the converse is not true, in general. Young gave an example of an Arens regular Banach algebra whose second dual is not Arens regular. Later Pym has polished Young's example for presenting more applicable examples. In this paper, we mimic the methods of Young and Pym to present examples of some Arens regular bilinear maps and module actions whose some extensions are not Arens regular.<br /> Finally, some relationships between the topological centers of certain Banach module actions are investigated.http://www.kjm-math.org/article_109816_130168186224906cc95a0197cf770c0e.pdf