Artinian ring

 ( Ring Theory ) 

In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes and rings that are finite-dimensional over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.

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 the left and right definitions coincide, but in general they are distinct from each other.

The Wedderburn–Artin theorem characterizes every simple ring Artinian ring as a Matrix ring 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 .

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.

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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., $\backslash mathbb\{Z\}/n\; \backslash mathbb\{Z\}$) is left and right Artinian.
• Let $k$ be a field. Then $kt/(t^n)$ is Artinian for every positive integer n.
• Similarly, $kx,y/(x^2,\; y^3,\; xy^2)\; =\; k\; \backslash oplus\; k\backslash cdot\; x\; \backslash oplus\; k\; \backslash cdot\; y\; \backslash oplus\; k\backslash cdot\; xy\; \backslash oplus\; k\; \backslash cdot\; y^2$ is an Artinian ring with maximal ideal $(x,y)$.
• Let $x$ be an endomorphism between a finite-dimensional vector space V. Then the subalgebra $A\; \backslash subset\; \backslash operatorname\{End\}(V)$ generated by $x$ is a commutative Artinian ring.
• If I is a zero ideal ideal of a Dedekind domain A, then $A/I$ is a principal Artinian ring.
• For each $n\; \backslash ge\; 1$, the full matrix ring $M\_n(R)$ 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 $x^\{n+1\}$ is (properly) contained in the ideal generated by $x^n$ for all natural numbers n. In contrast, if R is Noetherian so is Rx by the Hilbert basis theorem.
• The ring of integers $\backslash mathbb\{Z\}$ is a Noetherian ring but is not Artinian.

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Modules over Artinian rings Let M be a left module over a left Artinian ring. Then the following are equivalent (Hopkins' theorem): (i) M is finitely generated, (ii) M has finite length (i.e., has composition series), (iii) M is Noetherian, (iv) M is Artinian.

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Commutative Artinian rings Let A be a commutative Noetherian ring with unity. Then the following are equivalent.

• A is Artinian.
• A is a finite product of commutative Artinian local ring.
• is a semisimple ring, where nil( A) is the nilradical of A.
• Every finitely generated module over A has finite length. (see above)
• A has Krull dimension zero. (In particular, the nilradical is the Jacobson radical since are maximal.)
• $\backslash operatorname\{Spec\}A$ is finite and discrete.
• $\backslash operatorname\{Spec\}A$ is discrete.

Let k be a field and A a finitely generated k-algebra. Then A is Artinian if and only if A is finitely generated as a k-module.

An Artinian local ring is complete. A quotient ring and localization of an Artinian ring is Artinian.

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Simple Artinian ring One version of the Wedderburn–Artin theorem states that a simple ring Artinian ring A is a matrix ring over a division ring. Indeed, let I be a minimal (nonzero) right ideal of A, which exists since A is Artinian (and the rest of the proof does not use the fact that A is Artinian). Then, since $AI$ is a two-sided ideal, $AI\; =\; A$ since A is simple. Thus, we can choose $a\_i\; \backslash in\; A$ so that $1\; \backslash in\; a\_1\; I\; +\; \backslash cdots\; +\; a\_k\; I$. Assume k is minimal with respect to that property. Now consider the map of right A-modules:

$\backslash begin\{cases\}\; I^\{\backslash oplus\; k\}\; \backslash to\; A,\; \backslash \backslash \; (y\_1,\; \backslash dots,\; y\_k)\; \backslash mapsto\; a\_1y\_1\; +\; \backslash cdots\; +\; a\_k\; y\_k\; \backslash end\{cases\}$

This map is surjective, since the image is a right ideal and contains 1. If it is not injective, then, say, $a\_1y\_1\; =\; a\_2y\_2\; +\; \backslash cdots\; +\; a\_k\; y\_k$ with nonzero $y\_1$. Then, by the minimality of I, we have $y\_1\; A\; =\; I$. It follows:

$a\_1\; I\; =\; a\_1\; y\_1\; A\; \backslash subset\; a\_2\; I\; +\; \backslash cdots\; +\; a\_k\; I$,

which contradicts the minimality of k. Hence, $I^\{\backslash oplus\; k\}\; \backslash simeq\; A$ and thus $A\; \backslash simeq\; \backslash operatorname\{End\}\_A(A)\; \backslash simeq\; M\_k(\backslash operatorname\{End\}\_A(I))$.

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See also

• Artin algebra
• Artinian ideal
• Serial module
• Semiperfect ring
• Gorenstein ring
• Noetherian ring

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Citations

• (2025). 9783540353157, Springer-Verlag Berlin Heidelberg. ISBN 9783540353157
• Charles Hopkins. Rings with minimal condition for left ideals. Ann. of Math. (2) 40, (1939). 712–730.
• (2025). 9781852335878, Springer. ISBN 9781852335878
• (2025). 9783319086927, Springer. ISBN 9783319086927

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