An exact extension 0 -> A -> E -> B -> 0 which is not isomorphic to the direct or trivial extension A ⊕ B (or the semidirect/trivial splitting in the relevant category); equivalently, its class in the appropriate Ext or classification group is nonzero.

Principle

Non-split extensions represent genuine ways to assemble A and B into a new object that is not a direct sum; they correspond to nontrivial elements of extension groups and encode additional interaction data (extensions of modules, nontrivial group extensions, indecomposable objects).

Demonstration

A short exact sequence of abelian groups that does not admit a section yields a non-split extension: for instance, an extension class in Ext^1(B,A) that is nonzero produces an E not isomorphic to A ⊕ B; similarly, nontrivial group extensions produce groups that are not direct products.

Misapplication

Labeling every nonisomorphic middle object E as a non-split extension regardless of whether a splitting exists up to isomorphism; or confusing non-split with nonsimple or indecomposable, since a non-split extension may still decompose in other ways in a broader category.

Consequence

Non-split extensions create new algebraic structures and often give rise to indecomposable or genuinely new objects; their classification via Ext groups informs module theory, representation theory, and homological algebra by revealing hidden extension data.

Reversal

The reversal is a split extension: the exact sequence is isomorphic to the direct sum or admits a section and hence corresponds to the zero class in Ext; switching perspective highlights the role of a trivial extension class as the dividing line.

Boundary

Applies in abelian categories and in many nonabelian extension theories, but the meaning of 'split' depends on categorical context (existence of sections, semidirect product structures, or isomorphism to a product), so care is needed when moving between contexts.

Semantic Tension

Tension appears between the notion of 'non-split' and related notions like 'indecomposable' or 'nontrivial up to isomorphism'; while related, non-split specifically refers to failure to admit a splitting morphism or trivial class, not to all forms of nontriviality.

Synthesis

A non-split extension is an exact sequence whose extension class is nonzero, producing a middle object E that genuinely glues A and B together rather than decomposing as a direct sum; it is the algebraic witness to nontrivial gluing.