On Frobenius Extensions I.
On Frobenius Extensions I.
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DOI：
10.1017/s0027763000002075
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发表时间：
1960-08
影响因子：
0.8
通讯作者：
T. Nakayama;Tosiro Tsuzuku
中科院分区：
文献类型：
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作者：
T. Nakayama;Tosiro Tsuzuku
作为域上Frobenius代数概念的推广，Kasch[103]引入了环的Frobenius扩展。现在的作者[13]最近将Kasch的一个主要定理从它对地环的相当强的s环假设中解放出来。然而，即使去掉了s环对地环的假设，这个概念似乎还不够普遍，我们希望在本文及其后续文章中，在Frobenius推广的一个更普遍的概念的基础上发展这个理论。因此，我们将扩展的自由模性质替换为射影模性质（根据代数中的一般趋势），这实际上已经在Eilenberg和其中一位作者[4]的先前工作中对交换环上的Frobenius代数进行了处理，并且进一步将地环的自同构纳入Frobenius扩展的定义中（这在非交换环的情况下似乎很自然）。对于Frobenius扩展的这种广义概念，我们可以推广许多Kasch定理，包括那些经典Frobenius代数定理的直接扩展和那些本质上是新的定理，如上面提到的自同态环定理。在Hirata的[6]和Eilenberg-Nakayama[6]最近的论文中所发展的Frobenius扩展的同调性质也可以推广到我们现在的广义情况；在考虑内射维和弱维时，我们也会超过b[6]。
As a generalization of the notion of Frobenius algebras over a field Kasch [103 introduced that of Frobenius extensions of a ring. The present writers [13] recently freed one of Kasch’s main theorems from its rather strong S-ring assumption of the ground ring. However, even with the removal of the S-ring assumption of the ground ring the notion does not seem general enough, and we wish, in the present paper and its sequel, to develope the theory upon the basis of a more general notion of Frobenius extensions. Thus, we replace the free module property of the extension by the projective module property (according to a general tendency in algebra), which has been done in fact in case of Frobenius algebras over a commutative ring in a previous work by Eilenberg and one of the writers [4], and, further, take automorphisms of the ground ring into the definition of Frobenius extensions (which seems quite natural particularly in case of non-commutative rings). To such generalized notion of Frobenius extensions we may extend many of Kasch’s theorems, including those which are immediate extensions of classical theorems for Frobenius algebras and those which are essentially new, as the above alluded endomorphism ring theorem. Also homological properties of Frobenius extensions, as were developed in Hirata’s [6] recent paper in succession to Eilenberg-Nakayama [4], can be extended to our present generalized case; we shall also exceed [4], [6] somewhat in considering injective and weak dimensions.