On Properties of Product Acts over Monoids

Abstract

Let S be a monoid. It is known that flatness may be defined in a number of ways for the class of right S − acts, and these notions themselves naturally give rise to some weaker associated conditions. To date no work has been done on these properties of product acts. This article is concerned with closure under products of the class of right S − acts possessing one of these “flatness” properties and conversely, when these properties transfer from products to components. We consider such questions for both a general monoid S, and for monoids coming from some special classes. Specifically, we answer the question of when properties such as torsion freeness, principal weak flatness, GP − flatness, (weak) flatness, Conditions (P), (P′), etc. transfer from products of acts to their components. Also we give equivalences of when properties such as torsion freeness, Conditions (PWP), (P′) (EP), etc. transfer from acts to their products. Finally, we extend some results from Bulman-Fleming, S., Gilmour, A.

Key Words:

• Flatness properties
• Products
• S–act

2010 Mathematics Subject Classification:

• 20M30

ACKNOWLEDGMENTS

The authors would like to thank the referees for useful and valuable comments and suggestions relating to this article. They would also like to thank Professor V. Gould for providing the communication.

Notes

Communicated by V. Gould.