Artinian ring

Precisely, a ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other.

The Wedderburn–Artin theorem characterizes every simple Artinian ring as a ring of matrices over a division ring. This implies that a simple ring is left Artinian if and only if it is right Artinian.

The same definition and terminology can be applied to modules, with ideals replaced by submodules.

Although the descending chain condition appears dual to the ascending chain condition, in rings it is in fact the stronger condition. Specifically, a consequence of the Akizuki–Hopkins–Levitzki theorem is that a left (resp. right) Artinian ring is automatically a left (resp. right) Noetherian ring. This is not true for general modules; that is, an Artinian module need not be a Noetherian module.

Examples and counterexamples

• An integral domain is Artinian if and only if it is a field.
• A ring with finitely many, say left, ideals is left Artinian. In particular, a finite ring (e.g., ${\displaystyle \mathbb{Z}/n\mathbb{Z}}$) is left and right Artinian.
• Let ${\displaystyle k}$ be a field. Then ${\displaystyle k[t]/(t^n)}$ is Artinian for every positive integer n.
• Similarly, ${\displaystyle k[x,y]/(x^2,y^3,x{y}^2)=k\oplus k\cdot x\oplus k\cdot y\oplus k\cdot xy\oplus k\cdot y^2}$ is an Artinian ring with maximal ideal ${\displaystyle (x,y)}$.
• Let ${\displaystyle x}$ be an endomorphism between a finite-dimensional vector space V. Then the subalgebra ${\displaystyle A\subset \mathrm{End}(V)}$ generated by ${\displaystyle x}$ is a commutative Artinian ring.
• If I is a nonzero ideal of a Dedekind domain A, then ${\displaystyle A/I}$ is a principal Artinian ring.
• For each ${\displaystyle n\ge 1}$, the full matrix ring ${\displaystyle M_n(R)}$ over a left Artinian (resp. left Noetherian) ring R is left Artinian (resp. left Noetherian).

The following two are examples of non-Artinian rings.

• If R is any ring, then the polynomial ring R[x] is not Artinian, since the ideal generated by ${\displaystyle x^{n+1}}$ is (properly) contained in the ideal generated by ${\displaystyle x^n}$ for all natural numbers n. In contrast, if R is Noetherian so is R[x] by the Hilbert basis theorem.
• The ring of integers ${\displaystyle \mathbb{Z}}$ is a Noetherian ring but is not Artinian.