Goldie's theorem
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In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie (1920-2005) during the 1950s. It gives a result on the noetherian rings that have a classical ring of quotients, that is a semisimple artinian ring, and so of known structure by the Artin-Wedderburn theorem.
It does somewhat more, giving a precise characterisation of the situation by means of a definition. What is now termed a right Goldie ring is a ring R that is a module of finite rank as right module over itself, and satisfies the ascending chain condition on right annihilators of its elements r.
The statement of Goldie's theorem as now given is that the semiprime right Goldie rings are precisely those that have a right classical ring of quotients that is semisimple artinian.
[edit] References
- A. W. Goldie, The structure of prime rings under ascending chain conditions, Proc. London Math. Soc. 8 (1958) 589-608.
- A. W. Goldie, Semi-prime rings with maximal conditions, Proc. London Math. Soc. 10 (1960) 201-220.
