On ring structures under strong cup products

Can we construct a strong s-stage cohomology ring on inverse systems? The following statements show the answer to this question: After defining a strong cup product, we construct a ring structure with right identity under strong cup product and obtain a strong cup product preserving homomorphism induced by a rigid system map on the level of strong s-stage cochain groups. On the other hand, we show that the coboundary formula for strong s-stage cochains does not hold. Taylor & Francis Group, LLC.