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In ring theory, a branch of mathematics, a radical of a ring is an ideal of "bad" elements of the ring. The first example of a radical was the nilradical introduced in , based on a suggestion in . In the next few years several other radicals were discovered, of which the most important example is the Jacobson radical. The general theory of radicals was defined independently by and . ==Definitions== In the theory of radicals, rings are usually assumed to be associative, but need not be commutative and need not have an identity element. In particular, every ideal in a ring is also a ring. A radical class (also called radical property or just radical) is a class σ of rings possibly without identities, such that: (1) the homomorphic image of a ring in σ is also in σ (2) every ring ''R'' contains an ideal ''S''(''R'') in σ which contains every other ideal in σ (3) ''S''(''R''/''S''(''R'')) = 0. The ideal ''S''(''R'') is called the radical, or σ-radical, of ''R''. The study of such radicals is called torsion theory. For any class δ of rings, there is a smallest radical class ''L''δ containing it, called the lower radical of δ. The operator ''L'' is called the lower radical operator. A class of rings is called regular if every non-zero ideal of a ring in the class has a non-zero image in the class. For every regular class δ of rings, there is a largest radical class ''U''δ, called the upper radical of δ, having zero intersection with δ. The operator ''U'' is called the upper radical operator. A class of rings is called hereditary if every ideal of a ring in the class also belongs to the class. 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「Radical of a ring」の詳細全文を読む スポンサード リンク
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