More on S-pseudo radical of an ideal and a new class of ideals in noncommutative rings

We further develop the theory of the S-pseudo radical IS of ideals in noncommutative rings and introduce a new class of ideals, called rightSS-primary ideals. We establish the compatibility of IS with ring epimorphisms and provide explicit descriptions in idealization rings, direct products, and matrix rings. We also revisit S-principally-nilary ideals, relating them to elementwise and ideal-theoretic S-pseudo-radical conditions and obtaining equivalent characterizations under a finite-generation hypothesis. Finally, we study the behavior of right SS-primary ideals under standard ring constructions and compare them with right S-prime ideals. Examples show that these two notions are generally incomparable. They are equivalent for S-radical ideals, while in a fully idempotent ring, finite generation of KS ensures that every right SS-primary ideal K is right S-prime.