High Quality Content by WIKIPEDIA articles! Artinian ring, Ring , Emil Artin, Ascending chain condition, Finite ring, Vector space, Field ,Matrix ring, Division ring, Noetherian ring, Artin–Wedderburn theorem, Simple ring, Direct product, Nilradical of a ring, abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields.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 and for the two classes of rings mentioned above these concepts conincide, but in general they are different.The Artin–Wedderburn theorem characterizes all simple artinian rings as the matrix rings over a division ring.

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