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In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring ''R'' is a nonzero right ideal which contains no other nonzero right ideal. Likewise a minimal left ideal is a nonzero left ideal of ''R'' containing no other nonzero left ideals of ''R'', and a minimal ideal of ''R'' is a nonzero ideal containing no other nonzero two-sided ideal of ''R''. Said another way, minimal right ideals are minimal elements of the poset of nonzero right ideals of ''R'' ordered by inclusion. The reader is cautioned that outside of this context, some posets of ideals may admit the zero ideal, and so zero could potentially be a minimal element in that poset. This is the case for the poset of prime ideals of a ring, which may include the zero ideal as a minimal prime ideal. ==Definition== The definition of a minimal right ideal ''N'' of a module ''R'' is equivalent to the following conditions: *If ''K'' is a right ideal of ''R'' with ⊆ ''K'' ⊆ ''N'', then either ''K'' = or ''K'' = ''N''. *''N'' is a simple right ''R'' module. Minimal right ideals are the dual notion to the idea of maximal right ideals. 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「minimal ideal」の詳細全文を読む スポンサード リンク
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