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In abstract algebra, a nonzero ring ''R'' is a prime ring if for any two elements ''a'' and ''b'' of ''R'', ''arb = 0'' for all ''r'' in ''R'' implies that either ''a = 0'' or ''b = 0''. This definition can be regarded as a simultaneous generalization of both integral domains and simple rings. Prime ring can also refer to the subring of a field determined by its characteristic. For a characteristic 0 field, the prime ring is the integers, for a characteristic ''p'' field (with ''p'' a prime number) the prime ring is the finite field of order ''p'' (cf. prime field).〔Page 90 of 〕 ==Equivalent definitions== A ring ''R'' is prime if and only if the zero ideal is a prime ideal in the noncommutative sense. This being the case, the equivalent conditions for prime ideals yield the following equivalent conditions for ''R'' to be a prime ring: *For any two ideals ''A'' and ''B'' of ''R'', ''AB''= implies ''A''= or ''B''=. *For any two ''right'' ideals ''A'' and ''B'' of ''R'', ''AB''= implies ''A''= or ''B''=. *For any two ''left'' ideals ''A'' and ''B'' of ''R'', ''AB''= implies ''A''= or ''B''=. Using these conditions it can be checked that the following are equivalent to ''R'' being a prime ring: *All right ideals are faithful modules as right ''R'' modules. *All left ideals are faithful left ''R'' modules. 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「Prime ring」の詳細全文を読む スポンサード リンク
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