Study on Artinian rings with priority given to Quasi-Frobenius rings

(1)Oshio领队对拟Frobeniu环和Nakayama环作为Harada环的应用进行了广泛的研究(由Oshio引入)。在研究[1]中，Oshio对Harada环及其应用进行了综述，其中大部分是由于Oshio(2)在[2]、[3]中研究了扩张模直和为扩张模时的公开问题。我们准备好了这样一种情况，即这个问题应该通过考虑两种类型来研究。在本文中，我们通过引入广义相对内射性，完全解决了一类问题。最近，这种广义的相对内射性被称为“客观性”(Oshio Injectivity)(3)库拉米博士是首席研究员的学生。在[6]中，他研究了[3]的对偶结果，并给出了一个很好的结果。(4)Oshio[1]，[2]，[3]，Baba[4]，Koike[5]，Kurtom i[6]是本研究项目的重要论文。特别是，小池在文[5]中发现了一个不具有Nakayama自同构的拟Frobenius环的样例。这一结果是一个非常重要的结果，从某种意义上说，本文完成了首席研究员多年来对Artin环的研究。

(1) head investigator Oshiro has been extensively studied on Quasi-Frobeniu ring and Nakayama ring as applications of Harada rings (which was introduced by Oshiro). In research [1], Oshiro gave a survey on Harada ring and applications, most of which is due to Oshiro(2) In [2],[3], we studied the open problem when direct sum of extending modules are extending modules. We readied to a situation that this problem should be studied by considering two types. In this paper, we completely solved on one type, by introducing generalized relative injectivity. Recently, this generalized relative injectivity is called "objectivity" (Oshiro injectivity)(3)Dr Kuratomi is a student of the headinvestigator. In [6], he studied dual result of [3] and gave a very nice result.(4) Oshiro [1],[2],[3], Baba [4], Koike [5], Kuratomi [6] are important papers for this research project. In particular, in [5], Koike discovered an exsample of a Quasi-Frobenius ring without Nakayama automorphism. This result is a very important and , in some sense, this paper completes the headinvestigator's many years study on Artinian rings.