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In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem was first published by Thoralf Skolem in 1927 in his paper ''Zur Theorie der assoziativen Zahlensysteme'' (German: ''On the theory of associative number systems'') and later rediscovered by Emmy Noether. == Statement == In a general formulation, let ''A'' and ''B'' be simple unitary rings, and let ''k'' be the centre of ''B''. Notice that ''k'' is a field since given ''x'' nonzero in ''k'', the simplicity of ''B'' implies that the nonzero two-sided ideal ''BxB = (x)'' is the whole of ''B'', and hence that ''x'' is a unit. Suppose further that the dimension of ''B'' over ''k'' is finite, i.e. that ''B'' is a central simple algebra of finite dimension. Then given ''k''-algebra homomorphisms :''f'', ''g'' : ''A'' → ''B'', there exists a unit ''b'' in ''B'' such that for all ''a'' in ''A''〔Lorenz (2008) p.173〕 :''g''(''a'') = ''b'' · ''f''(''a'') · ''b''−1. In particular, every automorphism of a central simple ''k''-algebra is an inner automorphism.〔Gille & Szamuely (2006) p.40〕〔Lorenz (2008) p.174〕 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「Skolem–Noether theorem」の詳細全文を読む スポンサード リンク
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