Module over a quotient ring

Suppose $𝑉$ is a (left) module over $𝑅$ , and $𝔞 ⊴ 𝑅$ is a two-sided ideal. Then the quotient module $𝑉 / 𝔞 𝑉$ ¹ is canonically a module over the quotient ring $𝑅 / 𝔞$ with the action #m/thm/module

(𝑥 + 𝔞) (𝑣 + 𝔞 𝑉) = 𝑥 𝑣 + 𝔞 𝑉 .

Proof

First, note this is well-defined since for $𝑎 1, 𝑎 2 \in 𝔞$ and $𝑤 \in 𝑉$ ,

(𝑥 + 𝑎 1 + 𝔞) (𝑣 + 𝑎 2 + 𝔞 𝑉) = (𝑥 + 𝑎 1) (𝑣 + 𝑎 2 𝑤) + 𝔞 𝑉 = 𝑥 𝑣 + 𝑥 𝑎 2 𝑤 + 𝑎 1 𝑣 + 𝑎 1 𝑎 2 𝑤 ⏟____⏟____⏟ \in 𝔞 𝑉 + 𝔞 𝑉 = 𝑥 𝑣 + 𝔞 𝑉 .

The module axioms follow.

#state/tidy | #lang/en | #SemBr

Here $𝔞 𝑉$ denotes the submodule generated by elements of the form $𝑎 𝑣$ for $𝑎 \in 𝔞$ and $𝑣 \in 𝑉$ . ↩

Module over a quotient ring

Footnotes