Being an essential extension is transitive. Let M ⊆E ⊆E1 be submodules of a left R-module E1. If

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Being an essential extension is transitive. Let M ⊆E ⊆E1 be...

Being an essential extension is transitive. Let $M \subseteq E \subseteq E_{1}$ be submodules of a left $R$-module $E_{1}$. If $E$ is an essential extension of $M$ and $E_{1}$ is an essential extension of $E$, prove that $E_{1}$ is an essential extension of $M$.

Key Concepts

Module

A module is an algebraic structure similar to a vector space, but where the coefficients come from a ring instead of a field. It provides a framework for studying elements with operations of addition and scalar multiplication that satisfy certain axioms, and is fundamental in abstract algebra and homological algebra.

Submodule

A submodule is a subset of a module that is closed under the module operations of addition and scalar multiplication. Submodules inherit the structure of the parent module and are analogous to subspaces in vector spaces, allowing us to consider internal structures and decompositions within modules.

Essential Extension

An essential extension, or an essential submodule, is a concept in module theory where a module is said to be an essential extension of a submodule if every nonzero submodule of the extended module has a nonzero intersection with the submodule. This property ensures that the submodule is in a certain sense 'large' or 'dense' within the extension, playing a critical role in the analysis of module structure.

Transitivity of Essential Extensions

The transitivity property of essential extensions indicates that if one module is an essential extension of a submodule and a larger module is an essential extension of the intermediate module, then the larger module is an essential extension of the original submodule. This principle is pivotal in understanding how 'denseness' properties are maintained through successive module extensions.

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