On TNC-Rings

Abstract EN

This paper is a continuation of study rings relative to clean ring, where we study the concepts of 2-nil clean rings TN-Clear Ring (TNC-ring) of a noncommutative ring, which introduced by Chen and Sheibani.

these rings of ours naturally generalize the nil -clean rings.

we show that, if ??? is TNC - element, then ?(?−?)(?−?) is nilpotent and ?=?+?, where ?∈? and ?∈?(?), such that; ?(?−?)(?−?)=?, when every two pair of idempotent elements are orthogonal, and we determine condition under which the TNC-ring is strongly ?-Regular and we show that, if ? is commutative local ring and ? is maximal ideal.

then ? is TNC-ring if ?/?≅??.