|
In 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. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. 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 Artin–Wedderburn theorem characterizes all simple Artinian rings as the matrix rings over a division ring. This implies that a simple ring is left Artinian if and only if it is right Artinian. 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 == *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., ) is left and right Artinian. *Let ''k'' be a field. Then is Artinian for every positive integer ''n''. *If ''I'' is a nonzero ideal of a Dedekind domain ''A'', then is a principal Artinian ring.〔Theorem 459 of http://math.uga.edu/~pete/integral.pdf〕 *For each , the full matrix ring over a left Artinian (resp. left Noetherian) ring ''R'' is left Artinian (resp. left Noetherian). The ring of integers is a Noetherian ring but is not Artinian. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「artinian ring」の詳細全文を読む スポンサード リンク
|