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A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible. [5 ...
Module, in connection with modular decomposition of a graph, a kind of generalisation of graph components; Modularity (networks), a benefit function that measures the quality of a division of a Complex network into communities; Protein module or protein domain, a section of a protein with its own distinct conformation, often conserved in evolution
A free module is a module with a basis. [2] An immediate consequence of the second half of the definition is that the coefficients in the first half are unique for each element of M. If has invariant basis number, then by definition any two bases have the same cardinality. For example, nonzero commutative rings have invariant basis number.
A semisimple module M over a ring R can also be thought of as a ring homomorphism from R into the ring of abelian group endomorphisms of M. The image of this homomorphism is a semiprimitive ring, and every semiprimitive ring is isomorphic to such an image. The endomorphism ring of a semisimple module is not only semiprimitive, but also von ...
If M is a right R-module, then the set End R (M) of R-module endomorphisms is a ring with the multiplication given by composition. The endomorphism ring End R (M) acts on M by left multiplication defined by f.x = f(x). The bimodule property, that (f.x).r = f.(x.r), restates that f is a R-module homomorphism from M to itself.
Any module is the union of the directed set of its finitely generated submodules. A module M is finitely generated if and only if any increasing chain M i of submodules with union M stabilizes: i.e., there is some i such that M i = M. This fact with Zorn's lemma implies that every nonzero finitely generated module admits maximal submodules.
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(This is an equivalent definition since the tensor product is a right exact functor.) These definitions apply also if R is a non-commutative ring, and M is a left R-module; in this case, K, L and J must be right R-modules, and the tensor products are not R-modules in general, but only abelian groups.