On Artinian rings

在阿提宁环理论中，拟frobenius环（qf环）和Nakayama环（广义单列环）是重要的经典阿提宁环。在20世纪80年代初，Manabu Harada发现了一类新的artinian环，其中包含qf环和Nakayama环。现在这些新的环被称为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环和中山环具有森田自对偶。但最近有研究表明，原田环一般不具有森田自性。我们的第一个目标是给出这个问题的答案：什么样的原田环具有森田自二象性。具体地说，我们研究了成分类型的原田环。分量型的Harada环类包含了Nakayama环类，而Nakayama环被很好地表征为分量型的一种特殊的Harada环。此外，分量型的Harada环具有Morita弱对称自对偶性。在[Y。Baba and K. Oshiro: On a theorem of Fuller, Journal of Algebra, 154(1994), 86-94]，我们研究了著名的关于环上射射内射模的Fuller定理。T. Sumioka， M. Hoshino和M. Morimoto进一步研究了这项研究。另一方面，在[Y]。巴巴：拟射影模的注入性，投影。y的拟内射模，和内射模的射影覆盖，代数学报155(1994),415—434]，我们刻画了拟内射射影模和拟射影内射模。从这个角度出发，我们重新研究了上面的富勒定理。除了这些研究之外，我们还写了一篇题为“古典artinian rings and related topics”的讲义，其中包括上述结果和进一步未发表的结果。这本书将在几年内出版。

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"