If P is a cyclic projective R-module, show that P ≅e R for some idempotent e ∈R. Conclude that i

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If P is a cyclic projective R-module, show that P ≅e R for...

If $P$ is a cyclic projective $R$-module, show that $P \cong e R$ for some idempotent $e \in R$. Conclude that if $R$ has no idempotents other than 0 or 1 and if $V$ is an irreducible $R$-module with a projective cover, then $R$ is a local ring. Conversely, if $R$ is a local ring, show that all finitely generated $R$-modules have projective covers.

Key Concepts

Projective Cover

A projective cover of a module is a projective module that surjects onto the module in a minimal way, meaning that the kernel of the surjection is contained in the Jacobson radical of the projective module. Projective covers are useful in understanding and resolving modules by 'lifting' their structure to a projective setting, which often simplifies the study of their properties.

Irreducible (Simple) Module

An irreducible or simple module is a module that has no proper nonzero submodules. These modules form the building blocks of module theory, much like prime numbers in number theory, and are essential to the study of representation theory and in understanding the structure of more complex modules through Jordan–Hölder series or composition series.

Local Ring

A local ring is a ring that has a unique maximal ideal. Equivalently, in a local ring every element is either a unit or belongs to the unique maximal ideal. This concept is important in commutative algebra and algebraic geometry because local rings serve as models for the behavior of functions near a point, and they provide a natural setting for applying localization techniques.

Projective Module

A projective module is a module with the property that every surjective (onto) homomorphism onto it splits, meaning it can be viewed as a direct summand of a free module. This property makes projective modules flexible and easier to work with, particularly in constructing resolutions and studying extensions, which are central themes in homological algebra.

Cyclic Module

A cyclic module is a module that can be generated by a single element. This means that every element in the module can be written as the product of this generator with some element from the ring. The concept is fundamental in module theory because it simplifies the study of modules, providing a clear structure and often reducing problems to questions about the ring itself.

Idempotent Element

An idempotent element in a ring is an element e such that e² = e. Idempotents are significant because they can be used to decompose rings and modules into simpler components, often corresponding to direct summands. In the context of modules, the existence of nontrivial idempotents can signal a sort of decomposability or 'splitting' that influences the structure of projective modules.

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