A Novel Example of Non-Rigid Compatible with Ascending Chain Condition on Annihilators

Let R be an associative ring with unity. (Nasr-Isfahani,4102) proved that if R be an ACCPR-ring, : RR a rigid automorphism of R and  an  -derivation of R and R satisfies the ACC on right annihilators, then R[x;, ] is an ACCPR-ring. The main result in this note is an example on the ring of a compatible ring like S such that satisfies the ACC on right annihilators but is not ACCPR-ring. Also, we show that polynomial ring S[x] is not ACCPR-ring.

Keywords:

ACCPR-ring , Compatible ring , Rigid ring , ACC on annihilators