(i) Let M ⊆E be left R-modules. Prove that E is an essential...

(i) Let $M \subseteq E$ be left $R$-modules. Prove that $E$ is an essential extension of $M$ if and only if, for every nonzero $e \in E$, there is $r \in R$ with $r e \in M$ and $r e \neq 0$. (ii) Let $M \subseteq E$ be left $R$-modules, and let $\mathcal{S}$ be a chain of intermediate submodules; that is, $M \subseteq S \subseteq E$ for all $S \in \mathcal{S}$ and, if $S, S^{\prime} \in \mathcal{S}$, either $S \subseteq S^{\prime}$ or $S^{\prime} \subseteq S$. If each $S \in \mathcal{S}$ is an essential extension of $M$, use part (i) to prove that $\bigcup_{S \in \mathcal{S}} S$ is an essential extension of $M$.

(i) Let $M \subseteq E$ be left $R$-modules. Prove that $E$ is an essential extension of $M$ if and only if, for every nonzero $e \in E$, there is $r \in R$ with $r e \in M$ and $r e \neq 0$.
(ii) Let $M \subseteq E$ be left $R$-modules, and let $\mathcal{S}$ be a chain of intermediate submodules; that is, $M \subseteq S \subseteq E$ for all $S \in \mathcal{S}$ and, if $S, S^{\prime} \in \mathcal{S}$, either $S \subseteq S^{\prime}$ or $S^{\prime} \subseteq S$. If each $S \in \mathcal{S}$ is an essential extension of $M$, use part (i) to prove that $\bigcup_{S \in \mathcal{S}} S$ is an essential extension of $M$.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Chain of Submodules and Union

A chain of submodules refers to a collection of submodules that are comparable by inclusion; that is, for any two submodules in the chain, one is contained in the other. The union of such a chain forms a submodule, and properties like the essential extension of M can be extended to this union by leveraging the ordered structure of the chain. This method is commonly used in proofs that involve constructing larger modules from directed systems or chains of submodules.

Module Actions

The action of a ring R on a module is fundamental to module theory. In this context, for a nonzero element e in a module E, the existence of some r in R such that r · e is a nonzero element of a submodule M, establishes a link between e and M. This notion of ring elements 'transporting' nonzero elements into M is essential for characterizing the essential extension property.

Essential Extension

An essential extension of a module M is a module E that contains M as a submodule and has the property that every nonzero element in E has a nontrivial interaction with M. More precisely, any nonzero element of E, when acted on by some ring element from R, produces a nonzero element in M. This concept is crucial when considering how extensively M is embedded inside E.

Transcript

Please give Ace some feedback