On Rings Whose Skew Generalized Power Series Rings Are Left Ikeda–Nakayama

This paper continues our research on the Ikeda–Nakayama property (IN) in noncommutative ring constructions, building upon our previous works. Let be a ring, a strictly (or partially) ordered monoid, and a monoid homomorphism. The skew generalized power series ring consists of all functions from to whose support contains neither infinite descending chains nor infinite antichains, equipped with pointwise addition and multiplication defined by convolution twisted via the action of . Our study is closely connected to the question of whether a skew generalized power series ring over a left Ikeda–Nakayama ring is itself left Ikeda–Nakayama. This property is closely related to left self‐injective rings and provides insight into the characterization of such rings. We further examine their relationship with other important ring‐theoretic properties, such as being abelian, ‐Armendariz, and ‐McCoy.