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In mathematics, the characteristic of a ring ''R'', often denoted char(''R''), is defined to be the smallest number of times one must use the ring's multiplicative identity element (1) in a sum to get the additive identity element (0); the ring is said to have characteristic zero if this sum never reaches the additive identity. That is, char(''R'') is the smallest positive number ''n'' such that : for every element ''a'' of the ring (again, if ''n'' exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (''see ring''), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings. == Other equivalent characterizations == * The characteristic is the natural number ''n'' such that ''n''Z is the kernel of a ring homomorphism from Z to ''R''; * The characteristic is the natural number ''n'' such that ''R'' contains a subring ring homomorphism 0 will suffice, then the characteristic is 0. This is the right partial ordering because of such facts as that char ''A'' × ''B'' is the least common multiple of char ''A'' and char ''B'', and that no ring homomorphism ''ƒ'' : ''A'' → ''B'' exists unless char ''B'' divides char ''A''. * The characteristic of a ring ''R'' is ''n'' ∈ precisely if the statement ''ka'' = 0 for all ''a'' ∈ ''R'' implies ''n'' is a divisor of ''k''. The requirements of ring homomorphisms are such that there can be only one homomorphism from the ring of integers to any ring; in the language of category theory, Z is an initial object of the category of rings. Again this follows the convention that a ring has a multiplicative identity element (which is preserved by ring homomorphisms). 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「Characteristic (algebra)」の詳細全文を読む スポンサード リンク
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