Modules, submodules and exact sequences

An R-module is an abelian group with a compatible R-action. A submodule is closed under addition, negatives and every scalar action, so an ideal is a submodule of R viewed over itself. A list generates M when finite R-linear combinations cover M. A basis also requires unique coefficients, which need not be possible: the class of one generates Z/6Z but has the relation six times that class equals zero. The annihilator of an element records all scalars killing it; the annihilator of M kills every element simultaneously.

A presentation R^a→R^b→M→0 records relations among b chosen generators. The first map need not be injective because relations can themselves have relations. Over a field, finite-dimensional modules are free and dimension controls much of the story. Over a general ring, torsion and nonfree ideals survive. For example, an element killed by a nonzero integer cannot be part of an integer-module basis. Keeping the scalar ring explicit prevents an abelian group isomorphism from being mistaken for an isomorphism respecting a different ring action.

A sequence is exact at its middle module when the incoming image equals the outgoing kernel. A short exact sequence 0→N→M→Q→0 says that N embeds as the full kernel and every quotient class in Q is reached. It identifies Q with M/N, but does not choose representatives linearly. For a map f:M→N, the kernel is a submodule of the source, while the cokernel is N/im f and measures failure of surjectivity. Their roles differ even when both are represented by familiar cyclic groups in examples.

To split a short exact sequence, one needs a linear section Q→M of the quotient map, equivalently a compatible direct-sum decomposition M≅N⊕Q. Such a section exists for vector spaces but not for all modules. The sequence with integer multiplication by six followed by reduction modulo six is exact but not split: any homomorphism from Z/6Z into Z has zero image because Z has no nonzero six-torsion. A set-theoretic choice of residue representatives is not additive and cannot supply the missing module section.

If R is Noetherian, a submodule of a finite module is finite, so a finite module admits a presentation with finitely many relations. Without that hypothesis, finite generation of M does not guarantee finite generation of its relation module. This is one reason the next lesson emphasizes chain conditions instead of merely counting variables. The distinction between finite generation and finite presentation will later matter for flatness and local freeness. A surjection from a finite free module supplies generators, while controlling its kernel supplies a finite presentation.

Exact sequences make these issues transportable. Localization will preserve them, allowing a global relation problem to be studied near a prime. Tensor products always preserve their right-hand quotient behavior but can destroy injectivity unless flatness is present. Completion will preserve exactness for finite modules over Noetherian rings under the later stated hypotheses. Learning to identify the specific image and kernel at each arrow is therefore more useful than memorizing the shape of a diagram: every later operation must be checked against the exactness information it is supposed to preserve.