Group rings over dedekind domains
Group rings over dedekind domains
复制标题
dedekind 域上的组环
DOI：
10.1016/0021-8693(67)90045-2
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发表时间：
1967
影响因子：
0.9
通讯作者：
R. Larson
中科院分区：
文献类型：
--
作者：
R. Larson
设R是有单位的交换环，G是半群.众所周知，群环RG是R上的一个Hopf代数，其余积6：RG+ RG@ RG定义为6（g）= g@ g，g EG，增广E：RG-+ R定义为E（g）= 1，g EG.反之，当R上的一个Hopf代数A是一个群的群环时？若R是代数闭域，则A是半群的群环的充要条件由[3，定理3.21]给出.设R是整环，A是有限维自由R-模，[1]给出了A是群的群环的充要条件. 5]。本文给出了Dedekind整环上的一个Hopf代数是群的群环的条件，该Hopf代数是Dedekind整环上的一个非生成无挠模。设R是Dedekind整环，R上的一个Hopf代数是指单位元为无挠R-模的R-代数A，以及代数同态8：A-fA @ A和E：A+ R使得（106）s=（601）6，
Let R be a commutative ring with unit and G a semigroup. It is well known that the group ring RG is a Hopf algebra over R, with coproduct 6: RG+ RG@ RG defined by 6 (g)= g@ g, g EG, and augmentation E: RG-+ R defined by E (g)= 1, g E G. Conversely, when is a Hopf algebra A over R the group ring of a group? If R is an algebraically closed field, necessary and sufficient conditions for A to be the group ring of a semigroup are given by [3, Theorem 3.21. If R is an integral domain and A is a finitedimensional free R-module, necessary and sufficient conditions for A to be the group ring of a group are given in [. 5]. In this paper we give conditions for a Hopf algebra which is a finitely generated torsion-free module over a Dedekind domain to be the group ring of a group. If R is a Dedekind domain, by a Hopf algebra over R, we mean an R-algebra A with unit which is a finitely generated torsion-free R-module, together with algebra homomorphisms 8: A---f A@ A and E: A+ R such that (1 06) s=(60 1) 6 and