Model structures on modules over Ding-Chen rings
Model structures on modules over Ding-Chen rings
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DOI：
10.4310/hha.2010.v12.n1.a6
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发表时间：
2009-10
期刊：
影响因子：
--
通讯作者：
James Gillespie
中科院分区：
文献类型：
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作者：
James Gillespie
n-FC环是左、右凝聚环，其左、右自FP-内射维数均为n。Ding和Chen在{dingandchen 93}和{dingandchen 96}中的工作表明，这些环具有推广了$n$-Gorenstein环的性质.本文将具有有限（左和右）自FP-内射维度的（左和右）凝聚环称为Ding-Chen环。如果这个环是诺特环，那么它们就是戈伦斯坦环。我们看看类的模块，我们呼吁丁投射，丁内射和丁平这意味着类似于伊诺克' Gorenstein投射，Gorenstein内射和Gorenstein平坦的模块。我们开发这些模块的基本属性。然后，我们表明，每一个标准的模型结构的Mod-$R$，当$R$是一个Gorenstein环，推广到丁陈的情况下。证明了当R是交换Ding-Chen环，G是有限群时，群环R[G]是Ding-Chen环.
An $n$-FC ring is a left and right coherent ring whose left and right self FP-injective dimension is $n$. The work of Ding and Chen in \cite{ding and chen 93} and \cite{ding and chen 96} shows that these rings possess properties which generalize those of $n$-Gorenstein rings. In this paper we call a (left and right) coherent ring with finite (left and right) self FP-injective dimension a Ding-Chen ring. In case the ring is Noetherian these are exactly the Gorenstein rings. We look at classes of modules we call Ding projective, Ding injective and Ding flat which are meant as analogs to Enochs' Gorenstein projective, Gorenstein injective and Gorenstein flat modules. We develop basic properties of these modules. We then show that each of the standard model structures on Mod-$R$, when $R$ is a Gorenstein ring, generalizes to the Ding-Chen case. We show that when $R$ is a commutative Ding-Chen ring and $G$ is a finite group, the group ring $R[G]$ is a Ding-Chen ring.