强C4*-环的中心分裂、除稳定剩余与反环转移
Central Splitting, Division-Stable Residuals, and Opposite-Ring Transfer for Strongly C4*-Rings
AI总结：
本研究针对强C4*-环的结构问题，证明其中心分裂特性并分析分裂的不唯一性与并半格结构，给出反同构转移判据，丰富了环论中相关模类与环类的结构理论。
摘要：
已知的强C4*-模分解定理给出一个半单直和项与一个无直和方剩余直和项，当环绕模为投射模时二者满足双向Hom正交性。将该定理应用于正则模时，两个交叉角消失，因此定义该分解的幂等元是中心的。由此每个强右C4*-环可分裂为R≅Σ×T，其中Σ是半单阿廷环，T_T是无直和方的。\n 这种分裂未必唯一。我们将所有容许的中心分裂组织为一个并半格，并证明了一个比较定理：任意两个剩余因子有公共的直因子，且它们的补因子是除环的有限积。因此右到左的亏量与所选分裂无关。若中心幂等元满足升链条件，则存在唯一的最大容许幂等元；它包含所有中心半单阿廷直因子，从而产生一个不再含此类因子的典范剩余。域的无限积表明该有限性假设不能直接省略。\n 我们还证明了C4、C4*、半弱CS以及强C4*-模的反同构传递性。当剩余因子是半单无直和方环与自反核的积时，该性质可实现转移。该判据不假设正则性、交换性或左侧条件。斜洛朗环提供了非交换非正则的例子，而已知的单射非满射斜多项式构造给出了一个严格的单侧剩余障碍。
英文摘要：
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 \(R\cong Σ\times T\), where \(Σ\) 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.
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