artinian ring

• 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 $k[t]/(t^n)$ is Artinian for every positive integer n.
• Similarly, $k[x,y]/(x^2,\; y^3,\; xy^2)\; =\; k\; \backslash oplus\; k\backslash cdot\; x\; \backslash oplus\; k\; \backslash cdot\; y\; \backslash oplus\; k\backslash cdot\; xy\; \backslash oplus\; k\; \backslash cdot\; y^2$ 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 nonzero ideal of a Dedekind domain A, then $A/I$ is a principal Artinian ring.{{sfn|Clark|loc=Theorem 20.11|ps=none}}
• 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).{{sfn|Cohn|2003|loc=5.2 Exercise 11|ps=none}}

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 R[x] by the Hilbert basis theorem.
• The ring of integers $\backslash mathbb\{Z\}$ is a Noetherian ring but is not Artinian.

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.{{sfn|Bourbaki|2012|loc=VIII, p. 7|ps=none}}

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 rings.{{sfn|Atiyah|Macdonald|1969|loc=Theorems 8.7|ps=none}}
• {{nowrap|A / nil(A)}} is a semisimple ring, where nil(A) is the nilradical of A.{{citation needed|date=July 2012}}
• Every finitely generated module over A has finite length. (see above)
• A has Krull dimension zero.{{sfn|Atiyah|Macdonald|1969|loc=Theorems 8.5|ps=none}} (In particular, the nilradical is the Jacobson radical since prime ideals are maximal.)
• $\backslash operatorname\{Spec\}A$ is finite and discrete.
• $\backslash operatorname\{Spec\}A$ is discrete.{{sfn|Atiyah|Macdonald|1969|loc=Ch. 8, Exercise 2|ps=none}}

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 and localization of an Artinian ring is Artinian.

One version of the Wedderburn–Artin theorem states that a simple Artinian ring A is a matrix ring over a division ring. Indeed,{{sfn|Milnor|1971|p=144|ps=none}} 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 $A\; \backslash simeq\; \backslash operatorname\{End\}\_A(A)\; \backslash simeq\; M\_k(\backslash operatorname\{End\}\_A(I))$.

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

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• {{citation | last1=Auslander | first1=Maurice | last2=Reiten | first2=Idun | last3=Smalø | first3=Sverre O. | title=Representation theory of Artin algebras | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-41134-9 | doi=10.1017/CBO9780511623608 | mr=1314422 | year=1995 | volume=36 }}
• {{cite book |last1=Bourbaki |first1=Nicolas |year=2012 |title=Algèbre. Chapitre 8, Modules et anneaux semi-simples |publisher=Springer-Verlag Berlin Heidelberg |location=Heidelberg |isbn=978-3-540-35315-7 }}
• Charles Hopkins. Rings with minimal condition for left ideals. Ann. of Math. (2) 40, (1939). 712–730.
• {{citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969 }}
• {{cite book |last1=Cohn |first1=Paul Moritz |title=Basic algebra: groups, rings, and fields |year=2003 |publisher=Springer |isbn=978-1-85233-587-8 }}
• {{cite book |last1=Brešar|first1=Matej|title=Introduction to Noncommutative Algebra |year=2014 |publisher=Springer |isbn=978-3-319-08692-7 }}
• {{cite web | last1=Clark | first1=Pete L. | title=Commutative Algebra | url=http://math.uga.edu/~pete/integral.pdf | archive-url=https://web.archive.org/web/20101214214852/http://math.uga.edu/~pete/integral.pdf | archive-date=2010-12-14 }}
• {{citation | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=Princeton University Press | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72}}

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Category:Ring theory