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Polynomial Rings and Totally Ordered Monoid Rings

多项式环和全序幺半群环

基本信息

批准号：
13640025
负责人：
HIRANO Yasuyuki
金额：
$ 1.47万
依托单位：
OKAYAMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640025/
关键词：
ring module totally ordered monoid polynomial ring monoid ring annihilator ideal distributive ring differential operator モノイド順序半群零下イデアル平坦

项目摘要

1. As a generalization of a polynomial ring, we considered the monoid ring RG for a ring and a totally ordered G. Aring R is called a G-Armendarizring if the product of any two elements of RG is zero implies that the products of all of their coefficients are zero. We proved that this condition is equivalent to the bijectivify of the natural mapping between the set of left annihilator ideals of R and the set of that of RG. From this we know that if a G-Armendariz ring R is Baer then RG is Baer. We also introduced the concept of a G-quasi-Armendariz ring and gave a similar charactrization. We showed that if the left annihilator of any principal left ideal of R is a pure left ideal then R is G-quasi-Armendariz for any totally ordered monoid G. From this we know that any quasi-Baer ring is G-quasrArmendariz. Hence we proved that if R is' quasi-Baer then RG is quasi-Baer.2. Let I be an ideal I of a ring R. We considered when the annihilator of I in any left R-module M is a direct summaud of … More M. In other words, we considered when the preradical which assigns for any left R-module M the annihilator of I in M, is splitting. We showed that if an ideal I satisfies this condition and if R is I-torsion-free, then, for any ideal H containing I, R/H is a right hereditary right perfect ring. In particular, when R is commutative, we gave a necessary and sufficient condition for an ideal I to have this property. Moreover, as an application of a result of Osofsky and Smith, we proved that if all nonzero ideal I of a ring R have this property then any nonzero fector ring of R is a direct sum of prime rings.3. Let R be a ring and let U(R) denote the group of units in R. We consider R as a left U(R)-moduIe by the usual teft multiplication. We proved that the number of orbits is finite if and only if R is the direct sum of a finite ring and fuiitely many muserial rings. We also proved that if R has no nonzero finite fector ring, then this condition is equivalent to that R is a left Artinian left distributive ring.4. In 1981D. A. Jordan has shown the exsistance of a differential ring with no invertible derivation. In connection with this, we showed that under certain condition, a skew polynomial ring with n variables and n commutative derivations is D-simple for a derivation D. Less

1. 作为多项式环的推广，我们考虑了一个环和一个全序环的单弦环RG，如果RG中任意两个元素的积为零意味着它们的所有系数的积为零，则将RG称为g - armendarisring。证明了这个条件等价于R的左湮灭子理想集与RG的左湮灭子理想集之间的自然映射的对偶性。由此我们知道如果G-Armendariz环R是Baer那么RG也是Baer。我们还引入了g -拟armendariz环的概念，并给出了类似的刻画。我们证明了如果R的任何主左理想的左湮灭子是一个纯左理想，那么对于任何全序单群g， R就是g -拟曼达里兹，由此我们知道任何拟贝尔环都是g -拟曼达里兹。由此证明了如果R是准baer，则RG是准baer。设I是环r中的理想I，我们考虑当I在任意左r模M中的湮灭子是…更多M的直接和时，换句话说，我们考虑当给定任意左r模M中的I的湮灭子的前基是分裂的。我们证明了如果理想I满足这个条件并且R是I无扭的，那么对于任何包含I的理想H， R/H是一个右遗传的右完美环。特别地，当R是可交换的，我们给出了理想I具有这个性质的充分必要条件。此外，作为Osofsky和Smith的结果的应用，我们证明了如果环R的所有非零理想I都具有这个性质，则R的任何非零效应环都是素环的直接和。设R是一个环，U(R)表示R中的一组单元。我们用通常的左乘法把R看作一个左U(R)模。我们证明了轨道的数目是有限的当且仅当R是一个有限环与有限多个密环的直接和。我们还证明了如果R没有非零有限效应环，则这个条件等价于R是左阿蒂尼左分配环。在1981 d。a . Jordan证明了微分环没有可逆导数的存在性。为此，我们证明了在一定条件下，一个有n个变量和n个可交换导数的偏多项式环对于一个导数d - Less是D-simple的