Rings which have flat injective modules
Rings which have flat injective modules
复制标题
DOI：
10.1016/0021-8693(75)90049-6
复制
发表时间：
1975-06
影响因子：
0.9
通讯作者：
R. Colby
中科院分区：
文献类型：
--
作者：
R. Colby
我们称环R为左IF环，如果每个内射左R-模是平坦的。本文的目的是用几种方法刻画左IF环和双边IF环，并给出一些具有这种性质的环类的例子。由于拟Frobenius环的特征是每个内射左R-模都是投射模[S]，所以左完全左IF环就是拟Frobenius环。此外，任何正则环都是IF环[lo]。在[16]中简要地提到了IF环。第二节是主要定理。为了刻画左IF环，我们引入了两个比相干性弱的条件，我们称之为T-相干性和H-相干性（定义见下文）。我们得到了下列命题的等价性：（a）R是左IF;（B）R的每一个n-表示左R-模是自由模的子模;（c）R的特征模是平坦左R-模，R是右T-凝聚的;（d）R的n-生成右理想到R的每一个同态都可以延拓到R，R是右H-凝聚的.由于环R是拟Frobenius环当且仅当R是内射的且R是右Noether环，我们直接得到右Noether左IF环是拟Frobenius环.第二个主要结果是（双边）IF环被下列性质刻画：（a）R是左和右凝聚的，并且R到R中的任意一个R-生成的单边理想的同态都可以扩张到R;（B）R是左和右凝聚的，并且可以嵌入环S中，使得S ′和S ′是忠实平坦的和内射的。一个例子表明，IF属性不是左右对称的。在第三节中，我们证明了如果R是左IF环，则每个具有有限内射维数的左模是平坦的;如果R的弱整体维数是有限的，则R是正则的。我们还表明，Bezout域模一个非零主理想是一个IF环，并给出一个例子的IF环，这是不定期模其根。239
We shall call a ring R a left IF ring if every injective left R-module is flat. The purpose of this paper is to characterize left and two-sided IF rings in several ways and to give some examples of classes of rings which have this property. Since a quasi-Frobenius ring is characterized by the condition that every injective left R-module is projective [S], the left perfect left IF rings are precisely the quasi-Frobenius rings. Furthermore, any regular ring is an IF ring [lo]. IF rings are mentioned briefly in [16]. Section 2 contains the main theorems. In order to characterize left IF rings we introduce two conditions weaker than coherence which we call T-coherence and H-coherence(see below for definitions). We obtain the equivalence of the following statements:(a) R is left IF;(b) Every finitely presented left R-module is a submodule of a free module;(c) The character module of R, is a flat left R-module and R is right T-coherent;(d) Every homomorphism of a finitely generated right ideal of R into R can be extended to R and R is right H-coherent. Since a ring R is quasi-Frobenius if and only if R, is injective and R is right Noetherian, we obtain immediately that a right Noetherian left IF ring is quasi-Frobenius. The second main result is that (two-sided) IF rings are characterized by each of the following properties:(a) R is left and right coherent and every homomorphism of a finitely generated one-sided ideal of R into R can be extended to R; and (b) R is left and right coherent and can be embedded in a ring S such that $5’and S, are faithfully flat and injective. An example shows that the IF property is not left-right symmetric. In Section 3, we show that if R is a left IF ring then every left module which has finite injective dimension is flat and that if the weak global dimension of R is finite then R is regular. We also show that a Bezout domain modulo a nonzero principal ideal is an IF ring and give an example of an IF ring which is not regular modulo its radical. 239