How to check if a ring is Artinian?

Question

By definition, an Artinian ring is a ring that satisfies the descending chain condition on ideals. In practice, how to check if a ring is Artinian? For example, let $R$ be the quotient of the commutative ring $\mathbb{C}[x_1,x_2,x_3]$ the ideal $I$ generated by $x_1^2-1, x_2^2-1, x_3^2-1, (x_1-x_2)(x_1-x_3)$. How to check if $\mathbb{C}[x_1,x_2,x_3]/I$ is Artinian? Thank you very much.

Answer 1

Most examples I'm aware of in practice fall under the following two special cases.

Exercise 1: Every finite-dimensional algebra over a field $k$ is artinian.

Exercise 2: Every finite ring is artinian.

In this case, as Gunnar says in the comments, $R$ is a finite-dimensional algebra over $\mathbb{C}$. You can tell this because, after quotienting by $x_1^2 - 1, x_2^2 - 1, x_3^2 - 1$, the result is spanned by monomials where each $x_i$ occurs with an exponent of at most $1$, of which there are finitely many (in fact exactly $8$): $1, x_1, x_2, x_3, x_1 x_2, x_2 x_3, x_1 x_3, x_1 x_2 x_3$.