On Artinian rings

在Artin环的理论中，拟Frobenius环(QF-环)和Nakayama环(广义单列环)是重要的经典Artin环。20世纪80年代初，Manabu Harada发现了一类新的Artin环，它包含QF-环和Nakayama环。现在这些新的Artin环被称为Harada环，从Harada环的角度研究QF-环和Nakayama环是非常重要的。众所周知，QF-环和Nakayama环具有森田自对偶。但最近的研究表明，原田环一般不具有森田自对偶。我们的第一个目标是回答这个问题：什么样的原田环具有森田自对偶性。具体地说，我们研究了分支类型的原田环。分支类型的Harada环类包含Nakayama环的类，Nakayama环是分支类型的Harada环的一种特殊类型。此外，部件型的Harada环具有Morita弱对称自对偶.在[Y.Baba和K.Oshio：On a定理，Journal of Algebra 154(1994)，86-94]中，我们研究了著名的Fuller定理，它刻画了Artin环上的投射内射模.T.Sumioka，M.Hoshino和M.Morimoto进一步研究了这一研究。另一方面，在[Y.Baba：拟投射模的内射性，拟内射模的投射性，和内射模的投射覆盖，代数学报155(1994)，415--434]中，我们刻画了拟内射模和拟投射内射模。从这个角度出发，我们重新研究了上述富勒定理。除了这些研究外，我们还写了一篇题为《经典阿丁环及相关主题》的讲稿，其中包括上述结果和进一步未发表的结果。这本书将在几年后出版。

In the theory of artinian rings, quasi-Frobenius rings ( QF-rings) and Nakayama rings (generalized uniserial rings) are important classical artinian rings. And, in the early 1980s, Manabu Harada found a new class of artinian rings which contain QF-rings and Nakayama rings. Now these new artinian rings are called Harada rings and the study of QF-rings and Nakayama ring from the point of view of Harada rings is very important.It is well known that QF-rings and Nakayama rings have the Morita self-dualities. But recently it was showed that Harada rings do not have the Morita self-dialities in general. Our first aim is to give answers the question: What kind of Harada rings have the Morita self-dualities. To be specific, we study Harada rings of a component type. The class of Harada rings of a component type contains the class of Nakayama rings, and Nakayama rings are well characterized as a special kind of Harada rings of a component type. In addition, Harada rings of a component type have the Morita weakly symmetric self-dualities.In [Y. Baba and K. Oshiro: On a theorem of Fuller, Journal of Algebra 154 (1994), 86-94], we study the famous Fuller's theorem which characterizes projective injective modules over artinian rings. This reseach is further studied by T. Sumioka, M. Hoshino, and M. Morimoto. On the other hand, in [Y. Baba: Injectivity of quasi-projective modules, projectivit.y of quasi-injective modules, and projective cover of injective modules, Journal of Algebra 155 (1994), 415--434], we characterized quasi-injective projective modules and quasi-projective injective modules. From this point of view, we studied the above Fuller's Theorem again."In addition to these studies, we write a lecture note titled ""Classical artinian rings and related topics"" which includes the above results and further umpublished results. This book will be published in several years"