# Thread: finitely generated module & algebra

1. ## finitely generated module & algebra

Let $\displaystyle A$ be finitely generated as a $\displaystyle R$-algebra.

Then can I say $\displaystyle A$ is finitely generated as a $\displaystyle R$-module?

I cannot understand some part of definitions about finitely generated...

2. Originally Posted by Stiger
Let $\displaystyle A$ be finitely generated as a $\displaystyle R$-algebra.

Then can I say $\displaystyle A$ is finitely generated as a $\displaystyle R$-module?

I cannot understand some part of definitions about finitely generated...

The polynomial algebra $\displaystyle \mathbb{R}[x]$ is finitely generated as an $\displaystyle \mathbb{R}-$algebra, but it is NOT

a finitely generated $\displaystyle \mathbb{R}-$module...

** f.g. algebra = all the elements are finite sums of finite products of some elements in the algebra

** f.g. module = all the elements are finite sums AND PRODUCTS BY SCALARS of some elements in the module.

Tonio