# Artinian ring

2010 Mathematics Subject Classification: Primary: 16P20 Secondary: 17A [MSN][ZBL]

right Artinian ring

A ring that satisfies the minimum condition for right ideals, i.e. a ring in which any non-empty set $M$ of right ideals that is partially ordered by inclusion has a minimal element [1] — a right ideal from $M$ that does not strictly contain right ideals from $M$. In other words, an Artinian ring is a ring which is a right Artinian module over itself. A ring $A$ is an Artinian ring if and only if it satisfies the descending chain condition for right ideals, i.e. for any decreasing sequence of right ideals $B_1\supseteq B_2\supseteq\ldots$ there exists a natural number $m$ such that $B_m=B_{m+1}=\ldots$. The definition of a left Artinian ring is similar.

Each associative Artinian ring with a unit is right Noetherian (cf. Noetherian ring). Every finite-dimensional algebra over a field is an Artinian ring. The properties of Artinian rings in the class of alternative rings, and particularly in the class of associative rings, have been studied most fully (cf. Alternative rings and algebras; Associative rings and algebras). The Jacobson radical of an associative Artinian ring is nilpotent and contains every one-sided nil-ideal. A ring $A$ is a simple associative Artinian ring if and only if it is isomorphic with the ring of all matrices of some finite order over some associative skew-field. In the class of alternative rings any simple Artinian ring is either associative, or else is a Cayley–Dickson algebra over its centre, which is then a field. The structure of associative Artinian rings with zero Jacobson radical has been described (cf. Semi-simple ring). There exists a variant of this theorem in the case of alternative rings. For associative rings with non-zero Jacobson radical a fairly-advanced structure theory has been developed [1], [2]. Several classes of Artinian rings — quasi-Frobenius rings, uniserial rings, balanced rings (cf. Quasi-Frobenius ring; Uniserial ring; Balanced ring) — are being intensively studied.

#### References

 [1] E. Artin, C.J. Nesbitt, R.M. Thrall, "Rings with minimum condition" , Univ. Michigan Press , Ann Arbor (1946) [2] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) [3] Itogi Nauk. Algebra Topol. Geom. 1965 (1967) pp. 133–180 [4] Itogi Nauk. Algebra Topol. Geom. 1968 (1970) pp. 9–56