Central Splitting, Division-Stable Residuals, and Opposite-Ring Transfer for Strongly C4*-Rings

Abstract

The known decomposition theorem for strongly \(\Cfourstar\)-modules gives a semisimple summand and a summand-square-free residual summand, with two-way Hom-orthogonality when the ambient module is projective. Applied to the regular module, the two cross-corners vanish, so the idempotent defining the decomposition is central. Thus every strongly right \(\Cfourstar\)-ring splits as \(R\cong \Sigma\times T\), where \(\Sigma\) is semisimple artinian and \(T_T\) is summand-square-free. The splitting need not be unique. We organize all admissible central splittings into a join-semilattice and prove a comparison theorem: any two residual factors have a common direct factor, and their complementary factors are finite products of division rings. Consequently the right-to-left defect is independent of the chosen splitting. If the central idempotents satisfy the ascending chain condition, there is a unique greatest admissible idempotent; it captures every central semisimple artinian direct factor and hence yields a canonical residual with no further such factor. An infinite product of fields shows that this finiteness hypothesis cannot simply be omitted. We also prove anti-isomorphism transport for \(\Cfour\), \(\Cfourstar\), semi-weak-CS, and strongly \(\Cfourstar\) modules. It yields transfer whenever a residual factor is a product of a semisimple summand-square-free ring and a self-opposite core. This criterion does not assume regularity, exchange, or a left-sided hypothesis. Skew Laurent rings provide noncommutative nonregular examples, while the known injective-nonsurjective skew-polynomial construction gives a sharp one-sided residual obstruction.