THE POLARITY ALONG AN ELEMENT IN RINGS

Let R be a ring with unity 1. We introduce the definition of the (dual) polarity along an element in R. Given any a,d∈R, an element a is polar along d if there exists some p∈R such that p2=p∈comm(da), pd=d and da+1−p is a unit. Also, the dual polarity along an element is given. Simply polar elements, polar elements and well-supported elements are special cases of polar elements along d. Moreover, several properties and characterizations of the polarity along an element are investigated.