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In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative. Some standard references for Hensel rings are , , and . ==Definitions== In this article rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings. A local ring ''R'' with maximal ideal ''m'' is called Henselian if Hensel's lemma holds. This means that if ''P'' is a monic polynomial in ''R''(), then any factorization of its image ''P'' in (''R''/''m'')() into a product of coprime monic polynomials can be lifted to a factorization in ''R''(). A local ring is Henselian if and only if every finite ring extension is a product of local rings. A Henselian local ring is called strictly Henselian if its residue field is separably closed. A field with valuation is said to be Henselian if its valuation ring is Henselian. A ring is called Henselian if it is a direct product of a finite number of Henselian local rings. 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「Henselian ring」の詳細全文を読む スポンサード リンク
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