Varieties of Commutative Residuated Integral Pomonoids and Their Residuation Subreducts
Varieties of Commutative Residuated Integral Pomonoids and Their Residuation Subreducts
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交换剩余积分Pomonoid的变体及其剩余减约
DOI：
10.1006/jabr.1996.6834
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发表时间：
1997
期刊：
影响因子：
--
通讯作者：
J. Raftery
中科院分区：
文献类型：
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作者：
Willem J. Blok;J. Raftery
剩余的概念可以在Dedekin关于模的工作中找到，从那时起，它在理想理论中一直扮演着重要的角色。如果R是具有1的交换环，且I和J是R的理想，则I关于J的余数是通常表示的理想I：J，4定义为xgR：xJ：I。也就是说，I：J是由条件KJ：I当K：I：J刻画的理想。这个运算在R的所有理想的集合上定义得很好，并且用它的理想来捕捉环中的除法的概念。的确，如果R是整环且Ž。Ž.Ž.I，j g R是这样的：对于某个元素k g R，i S kj，则i：j S k；Ž。Krull Kru24和Wx Ward和Dilworth WD39开始了一系列的研究，表明诺特环的许多结构理论可以在抽象的格子环境中通过适当的乘法运算Ž得到。Ž是从理想乘法和余数中抽象出来的。如上所述的理想残留物。
The notion of residuation can already be found in Dedekind’s work on modules, and it has played an important role in ideal theory ever since. If R is a commutative ring with 1, and I and J are ideals of R then the residual of I with respect to J is the ideal commonly denoted I: J and 4 defined as x g R: xJ : I . That is, I: J is the ideal characterized by the condition KJ : I iff K : I: J. This operation is well defined on the collection of all ideals of R and serves to capture the concept of division in the ring in terms of its ideals. Indeed, if R is an integral domain and Ž . Ž . Ž . i, j g R are such that i s kj for some element k g R, then i : j s k ; Ž . w x for r g R, r stands here for the ideal generated by r. Krull Kru24 and w x Ward and Dilworth WD39 started a long line of investigation showing that much of the structure theory of Noetherian rings can be obtained in the abstract setting of lattices with a suitable multiplication operation Ž . Ž abstracted from ideal multiplication and residuation abstracted from . ideal residuation as described above .