Аннотация:We examine the problem of the linearity of an algebraic torus action in the associativesetting. We prove the free algebra analog of a classical theorem of Białynicki-Birula,which establishes linearity of maximal torus action. Additionally, we formulate andprove linearity theorems for specific classes of regular actions, and provide a framework for constructing non-linearizable actions, analogous to the work of Asanuma.This framework has applications in the study of the Associative Cancellation Conjecture. Furthermore, we show the existence of two non-isomorphic algebras, whose freeproducts with a polynomial ring are isomorphic.Keywords Torus action · Linearity · Free algebra · Automorphism liftingMathematics Subject Classification 16S10 · 14R10 · 14R20
