A Study of Involution of Prime Rings with Commutativity and Left Centralizers

Exploring the Relationship between Involution and Commutativity in Prime Rings

Abstract

In the present study we give a brief exposition of some important terminology in the theory ofrings and algebras. Examples and counter examples are also included in this study to make the matterpresented in the study self-explanatory and to give a clear sketch of the various notions.In the earlystages of general ring theory, striking success of that theory were theorems which asserted thecommutativity of the ring when the elements of a ring were subjected to certain types of algebraicconditions.A good cross-section of such results, and the techniques needed to obtain them, can befound where further references can be found. Later as the theory evolved, many authors investigatedthe relationship between the commutativity of the ring R and certain special types of maps on R. In thisdirection the concept of centralizing and commuting maps is of great importance. A mapping f of R intoitself is called centralizing if [f(x), x] ∈ Z(R) holds for all x ∈ R in the special case when [f(x), x] = 0 holdsfor all x ∈ R, the mapping f is said to be commuting.