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

(1)首席研究员Oshiro广泛研究了准frobeniu环和Nakayama环作为原田环（由Oshiro介绍）的应用。在[1]中，Oshiro对Harada环及其应用进行了综述，其中大部分是由Oshiro(2)完成的。在[2]，[3]中，我们研究了可拓模的直接和为可拓模时的开放问题。我们准备考虑两种情况来研究这个问题。本文通过引入广义相对注入性，完全解决了一类问题。最近，这种广义的相对注入被称为“客观性”（Oshiro注入）。(3)Kuratomi博士是首席研究员的学生。在b[6]中，他研究了[3]的对偶结果，并给出了一个很好的结果。(4) Oshiro [1],[2],[3], Baba [4], Koike [5]， Kuratomi[6]是本课题的重要论文。特别是在2010年，Koike发现了一个没有中山自同构的拟frobenius环的例子。这一结果非常重要，从某种意义上说，这篇论文完成了首席研究员多年来对阿提尼亚环的研究。

(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.