Some Results about Acts over Monoid and Bounded Linear Operators

Abstract

This study delves into the properties of the associated act V over the monoid S of sinshT. It examines the relationship between faithful, finitely generated, and separated acts, as well as their connections to one-to-one and onto operators. Additionally, the correlation between acts over a monoid and modules over a ring is explored. Specifically, it is established that  functions as an act over S if and only if  functions as module, where T represents a nilpotent operator. Furthermore, it is proved that when T is onto operator and  is finitely generated, is guaranteed to be finite-dimensional. Prove that for any bounded operator the following,  is acting over S if and only if  is a module where T is a nilpotent operator, is a faithful act over S, where T is any bounded linear operator, if  T is any bounded operator, then   is separated,   if  is separated act over S, Then T  is injective, if a basis K = {v_j, j} for V, then every element w of  can be composed as =(p_n (T) + . v, for some v in V, and  put T as similar to any operator  from  to  and V as a finite dimensional normed space, then  is Noetherian act over S if S is Noetherian.

Received 23/05/2023

Revised 10/09/2023

Accepted 12/09/2023

Published Online First 20/02/2024