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In mathematics, especially ring theory, a regular ideal can refer to multiple concepts. In operator theory, a right ideal in a (possibly) non-unital ring ''A'' is said to be regular (or modular) if there exists an element ''e'' in ''A'' such that for every . In commutative algebra a regular ideal refers to an ideal containing a non-zero divisor.〔Non-zerodivisors in commutative rings are called ''regular elements''.〕 This article will use "regular element ideal" to help distinguish this type of ideal. A two-sided ideal of a ring ''R'' can also be called a (von Neumann) regular ideal if for each element ''x'' of there exists a ''y'' in such that ''xyx''=''x''. Finally, regular ideal has been used to refer to an ideal ''J'' of a ring ''R'' such that the quotient ring ''R''/''J'' is von Neumann regular ring.〔Burton, D.M. (1970) ``A first course in rings and ideals.'' Addison-Wesley. Reading, Massachusetts .〕 This article will use "quotient von Neumann regular" to refer to this type of regular ideal. Since the adjective ''regular'' has been overloaded, this article adopts the alternative adjectives ''modular'', ''regular element'' ''von Neumann regular'', and ''quotient von Neumann regular'' to distinguish between concepts. ==Properties and examples== 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「regular ideal」の詳細全文を読む スポンサード リンク
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