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In mathematics, an element, ''x'', of a ring, ''R'', is called nilpotent if there exists some positive integer, ''n'', such that ''x''''n'' = 0. The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras.〔Polcino Milies & Sehgal (2002), ''An Introduction to Group Rings''. p. 127.〕 == Examples == *This definition can be applied in particular to square matrices. The matrix :: :is nilpotent because ''A''3 = 0. See nilpotent matrix for more. * In the factor ring Z/9Z, the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9. * Assume that two elements ''a'', ''b'' in a (non-commutative) ring ''R'' satisfy ''ab'' = 0. Then the element ''c'' = ''ba'' is nilpotent (if non-zero) as ''c''2 = (''ba'')2 = ''b''(''ab'')''a'' = 0. An example with matrices (for ''a'', ''b''): :: : Here ''AB'' = 0, ''BA'' = ''B''. *The ring of coquaternions contains a cone of nilpotents. 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「nilpotent」の詳細全文を読む スポンサード リンク
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