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In mathematics, the Köthe conjecture is a problem in ring theory, open . It is formulated in various ways. Suppose that ''R'' is a ring. One way to state the conjecture is that if ''R'' has no nil ideal, other than , then it has no nil one-sided ideal, other than . This question was posed in 1930 by Gottfried Köthe (1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings〔John C. McConnell, James Christopher Robson, Lance W. Small , ''Noncommutative Noetherian rings'' (2001), p. 484.〕 and right Noetherian rings,〔Lam, T.Y., ''A First Course in Noncommutative Rings'' (2001), p.164.〕 but a general solution remains elusive. ==Equivalent formulations== The conjecture has several different formulations:〔Krempa, J., “Logical connections between some open problems concerning nil rings,” ''Fundamenta Mathematicae'' 76 (1972), no. 2, 121–130.〕〔Lam, T.Y., ''A First Course in Noncommutative Rings'' (2001), p.171.〕〔Lam, T.Y., ''Exercises in Classical Ring Theory'' (2003), p. 160.〕 # (Köthe conjecture) In any ring, the sum of two nil left ideals is nil. # In any ring, the sum of two one-sided nil ideals is nil. # In any ring, every nil left or right ideal of the ring is contained in the upper nil radical of the ring. # For any ring ''R'' and for any nil ideal ''J'' of ''R'', then the matrix ideal M''n''(''J'') is a nil ideal of M''n''(''R'') for every ''n''. # For any ring ''R'' and for any nil ideal ''J'' of ''R'', then the matrix ideal M''2''(''J'') is a nil ideal of M''2''(''R''). # For any ring ''R'', the upper nilradical of M''n''(''R'') is the set of matrices with entries from the upper nilradical of ''R'' for every positive integer ''n''. # For any ring ''R'' and for any nil ideal ''J'' of ''R'', the polynomials with indeterminate ''x'' and coefficients from ''J'' lie in the Jacobson radical of the polynomial ring ''R''(). # For any ring ''R'', the Jacobson radical of ''R''() consists of the polynomials with coefficients from the upper nilradical of ''R''. 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「Köthe conjecture」の詳細全文を読む スポンサード リンク
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