|
In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals. For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form where ''n'' is a square-free integer. So, is a semiprime ideal of the integers, but is not. The class of semiprime rings includes semiprimitive rings, prime rings and reduced rings. Most definitions and assertions in this article appear in and . ==Definitions== For a commutative ring ''R'', a proper ideal ''A'' is a semiprime ideal if ''A'' satisfies either of the following equivalent conditions: * If ''x''''k'' is in ''A'' for some positive integer ''k'' and element ''x'' of ''R'', then ''x'' is in ''A''. * If ''y'' is in ''R'' but not in ''A'', all positive integer powers of ''y'' are not in ''A''. The latter condition that the complement is "closed under powers" is analogous to the fact that complements of prime ideals are closed under multiplication. As with prime ideals, this is extended to noncommutative rings "ideal-wise". The following conditions are equivalent definitions for a semiprime ideal ''A'' in a ring ''R'': * For any ideal ''J'' of ''R'', if ''J''''k''⊆''A'' for a positive natural number ''k'', then ''J''⊆''A''. * For any ''right'' ideal ''J'' of ''R'', if ''J''''k''⊆''A'' for a positive natural number ''k'', then ''J''⊆''A''. * For any ''left'' ideal ''J'' of ''R'', if ''J''''k''⊆''A'' for a positive natural number ''k'', then ''J''⊆''A''. * For any ''x'' in ''R'', if ''xRx''⊆''A'', then ''x'' is in ''A''. Here again, there is a noncommutative analogue of prime ideals as complements of m-systems. A nonempty subset ''S'' of a ring ''R'' is called an n-system if for any ''s'' in ''S'', there exists an ''r'' in ''R'' such that ''srs'' is in ''S''. With this notion, an additional equivalent point may be added to the above list: * ''R''\''A'' is an n-system. The ring ''R'' is called a semiprime ring if the zero ideal is a semiprime ideal. In the commutative case, this is equivalent to ''R'' being a reduced ring, since ''R'' has no nonzero nilpotent elements. In the noncommutative case, the ring merely has no nonzero nilpotent right ideals. So while a reduced ring is always semiprime, the converse is not true.〔The full ring of two-by-two matrices over a field is semiprime with nonzero nilpotent elements.〕 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「Semiprime ring」の詳細全文を読む スポンサード リンク
|