クルルの単項イデアル定理




可換環論（次元論）において、クルルの単項イデアル定理（英: Krull's principal ideal theorem, Krulls Hauptidealsatz）は、ネーター環の素イデアルの高さについての基本的な定理である。



ステートメント


ネーター環 A の単項イデアル I の極小素因子（I を含む極小素イデアル）の高さは 1 以下である。とくに、x を A の零因子でも単元でもない元とすると、x を含む極小素イデアルの高さは 1 である。



一般化


次の定理はクルルの高度定理（英: Krull's height theorem）と呼ばれる。


ネーター環の r 個の元で生成されるイデアルの極小素因子の高さは高々 r である。


参考文献




  • Hartshorne, Robin (1977), Algebraic Geometry（英語版）, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
    0463157     

  • 松村英之 『可換環論』 共立出版、東京、1980年。
    ISBN 4-320-01658-0。

  • 堀田, 良之 『可換環と体』 岩波書店、2006年。
    ISBN 4-00-005198-9。




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