Exact-Sequence Stability and Ambient Realizations for $C4^{\ast}$-Modules

TL;DR

This paper develops an exact-sequence framework for C4*-modules, identifying conditions for their stability under extensions and submodules, and explores ambient categories where these modules form extension-closed classes.

Contribution

It introduces explicit hypotheses ensuring the stability of C4*-modules under various exact sequence operations and categorifies their ambient settings.

Findings

Identifies conditions for C4*-modules to be stable under split extensions and kernels.

Establishes obstruction patterns for extension stability and submodule passage.

Provides ambient verification theorems for semisimple modules over rings.

Abstract

The theory of C4*-modules is presently dominated by decomposition methods, but it lacks a systematic closure theory. In particular, it is not known in general whether the C4* property is preserved under extensions, kernels, cokernels, or short exact sequences. This is a structural difficulty, since C4*-type conditions are governed by summand behavior and comparison of submodules, and such data are not automatically respected by exact sequences. This paper develops an exact-sequence framework for C4*-modules and strongly C4*-modules. It identifies explicit hypotheses under which these classes are stable under split extensions, admissible kernels, admissible cokernels, and short exact extensions. The paper also separates positive and negative directions: closure results are established under summand-lifting and factor-control assumptions, while converse results show that these hypotheses cannot in general be removed. This produces concrete obstruction patterns for extension stability and for passage to submodules and factor modules. A further aim is categorical. Natural ambient settings are identified in which C4*-modules form an extension-closed subcategory, or at least a relative exact class appropriate to summand-sensitive module theory. Finally, concrete ambient verification theorems are proved: semisimple right modules over any ring provide a canonical exact environment, and over a semisimple artinian ring this extends to the full module category.