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Indecomposable module
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.
Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple means "no proper submodule" N < M, while indecomposable "not expressible as N ⊕ P = M".
A direct sum of indecomposables is called completely decomposable;[citation needed] this is weaker than being semisimple, which is a direct sum of simple modules.
A direct sum decomposition of a module into indecomposable modules is called an indecomposable decomposition.
In many situations, all modules of interest are completely decomposable; the indecomposable modules can then be thought of as the "basic building blocks", the only objects that need to be studied. This is the case for modules over a field or PID, and underlies Jordan normal form of operators.
Modules over fields are vector spaces. A vector space is indecomposable if and only if its dimension is 1. So every vector space is completely decomposable (indeed, semisimple), with infinitely many summands if the dimension is infinite.
Finitely-generated modules over principal ideal domains (PIDs) are classified by the structure theorem for finitely generated modules over a principal ideal domain: the primary decomposition is a decomposition into indecomposable modules, so every finitely-generated module over a PID is completely decomposable.
Explicitly, the modules of the form R/pn for prime ideals p (including p = 0, which yields R) are indecomposable. Every finitely-generated R-module is a direct sum of these. Note that this is simple if and only if n = 1 (or p = 0); for example, the cyclic group of order 4, Z/4, is indecomposable but not simple – it has the subgroup 2Z/4 of order 2, but this does not have a complement.
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Indecomposable module
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.
Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple means "no proper submodule" N < M, while indecomposable "not expressible as N ⊕ P = M".
A direct sum of indecomposables is called completely decomposable;[citation needed] this is weaker than being semisimple, which is a direct sum of simple modules.
A direct sum decomposition of a module into indecomposable modules is called an indecomposable decomposition.
In many situations, all modules of interest are completely decomposable; the indecomposable modules can then be thought of as the "basic building blocks", the only objects that need to be studied. This is the case for modules over a field or PID, and underlies Jordan normal form of operators.
Modules over fields are vector spaces. A vector space is indecomposable if and only if its dimension is 1. So every vector space is completely decomposable (indeed, semisimple), with infinitely many summands if the dimension is infinite.
Finitely-generated modules over principal ideal domains (PIDs) are classified by the structure theorem for finitely generated modules over a principal ideal domain: the primary decomposition is a decomposition into indecomposable modules, so every finitely-generated module over a PID is completely decomposable.
Explicitly, the modules of the form R/pn for prime ideals p (including p = 0, which yields R) are indecomposable. Every finitely-generated R-module is a direct sum of these. Note that this is simple if and only if n = 1 (or p = 0); for example, the cyclic group of order 4, Z/4, is indecomposable but not simple – it has the subgroup 2Z/4 of order 2, but this does not have a complement.