Why is $\mathbb{R}[X_1, X_2, \dots]$ a Goldie ring.

Question

On the database of ring theory, I read that

$$\mathbb{R}[X_1, X_2, \dots]$$

is a Goldie ring. Is there an easy way to show that all chains of left annihilators and direct sums stabilize? This seems odd to me as this ring is clearly not Noetherian since we have the strict chain

$$(X_1) \subseteq (X_1, X_2) \subseteq (X_1, X_2, X_3) \subseteq \dots$$

Answer 1

The uniform dimension of a commutative domain is obviously $1$, as $R_R$ and $_RR$ are both uniform modules. When the ring is not commutative, you can have infinite uniform dimension.

Every domain, commutative or not, has two annihilator ideals, so the condition on annihilator always holds in that case.

Answer 2

That ring is a commutative domain, so is an Ore domain. A result of Goldie (Thm. 10.22 in Lam Lectures on Modules and Rings) tells you that the uniform dimension of the ring is finite. Unless I'm missing something, the stabilisation on left annihilators is trivial because the ring is a domain.