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In commutative ring theory, a branch of mathematics, the radical of an ideal ''I'' is an ideal such that an element ''x'' is in the radical if some power of ''x'' is in ''I''. A radical ideal (or semiprime ideal) is an ideal that is its own radical (this can be phrased as being a fixed point of an operation on ideals called 'radicalization'). The radical of a primary ideal is prime. Radical ideals defined here are generalized to noncommutative rings in the Semiprime ring article. ==Definition== The radical of an ideal ''I'' in a commutative ring ''R'', denoted by Rad(''I'') or , is defined as : Intuitively, one can think of the radical of ''I'' as obtained by taking all the possible roots of elements of ''I''. Equivalently, the radical of ''I'' is the pre-image of the ideal of nilpotent elements (called nilradical) in .〔A direct proof can be give as follows: Let ''a'' and ''b'' be in the radical of an ideal ''I''. Then, for some positive integers ''m'' and ''n'', ''a''''n'' and ''b''''m'' are in ''I''. We will show that ''a'' + ''b'' is in the radical of ''I''. Use the binomial theorem to expand (''a''+''b'')''n''+''m''−1 (with commutativity assumed): : For each ''i'', exactly one of the following conditions will hold: *''i'' ≥ ''n'' *''n'' + ''m'' − 1 − ''i'' ≥ ''m''. This says that in each expression ''a''''i''''b''''n''+''m''− 1 − ''i'', either the exponent of ''a'' will be large enough to make this power of ''a'' be in ''I'', or the exponent of ''b'' will be large enough to make this power of ''b'' be in ''I''. Since the product of an element in ''I'' with an element in ''R'' is in ''I'' (as ''I'' is an ideal), this product expression will be in ''I'', and then (''a''+''b'')''n''+''m''−1 is in ''I'', therefore ''a''+''b'' is in the radical of ''I''. To finish checking that the radical is an ideal, we take an element ''a'' in the radical, with ''a''''n'' in ''I'' and an arbitrary element ''r''∈''R''. Then, (''ra'')''n'' = ''r''''n''''a''''n'' is in ''I'', so ''ra'' is in the radical. Thus the radical is an ideal.〕 The latter shows is an ideal itself, containing ''I''. If the radical of ''I'' is finitely generated, then some power of is contained in ''I''. In particular, If ''I'' and ''J'' are ideals of a noetherian ring, then ''I'' and ''J'' have the same radical if and only if ''I'' contains some power of ''J'' and ''J'' contains some power of ''I''. If an ideal ''I'' coincides with its own radical, then ''I'' is called a ''radical ideal'' or ''semiprime ideal''. 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 an ideal」の詳細全文を読む スポンサード リンク
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