Research on ring theory relative to self-injective rings

1.拟Harada环的结构与Morita对偶我们研究了拟Harada环的结构与Morita对偶，得到了许多结果。与Harada环一样，我们证明了每一个拟Harada环都是由QF环构造的。也就是说，从一个QF环开始，继续取某个子环的一个因子环（我们称这样的环为对角完备环）。然后我们可以到达任何准原田环。我们还研究了好的自对偶性，这是自对偶性的一种特殊类型。作为拟Harada环的结构结果的应用，我们证明了以下结果：每个局部分配右序列环具有良好的自对偶性，每个局部分配右QF-2环具有几乎自对偶性，这是自对偶性的推广.这些结果部分地回答了Azumaya猜想，即每一个正合Artin环都有一个自对偶。我们还改进了Y.巴巴关于自对偶的Auslander环的系列环。2.环扩张与Morita对偶B. J. Muller证明了具有Morita对偶的环A的环扩张R也具有Morita对偶，如果R满足一定的条件。证明了如果两个环A和B是Morita对偶的，则某些A-环和Bring的范畴是范畴等价的，相应的A-环和Bring是Morita对偶的.这是对Muller结果的改进。当R是A的有限中心化扩张且作为A-模是自由的时，我们还确定了B和S之间的关系，其中A和B是Morita对偶，R和S是Morita对偶.这个结果统一和推广了Mano关于有限中心扩张的自对偶的一个定理和Haack-Fuller关于半群环的Morita对偶的一个定理。

1. Structure of quasi-Harada rings and Morita dualityWe investigated structure of quasi-Harada rings and Morita duality and obtained many results. As well as the case of Harada rings, we proved that every quasi-Harada ring is constructed from a QF ring. That is, start with a QF ring and continue to take a factor ring of a certain subring (we named such a ring a diagonally complete ring). Then we can reach any quasi-Harada ring. We also studied good self-duality, a special type of self-duality. As applications of the result about structure of quasi-Harada rings, we proved the following every locally distributive right serial ring has a good self-duality and every locally distributive right QF-2 ring has almost self duality, a generalization of self-duality. These results are partial answers of Azumaya's conjecture, which states that every exact artinian ring has a self duality. We also improved a recent result of Y. Baba about self-duality of Auslander rings of serial rings.2. Ring extensions and Morita dualityB.J.Muller proved that a ring extension R of a ring A with Morita duality also has a Morita duality if R satisfies some condition. We proved that if two rings A and B are Morita dual, then categories of certain A-rings and Brings are category equivalent and corresponding A-ring and Bring are Morita dual. This is an improvement of Muller's result. We also determined a relation between B and S in case R is a finite centralizing extension of A and is free as an A-module, where A and B are Morita dual and R and S are Morita dual. This result unifies and generalizes a theorem of Mano about self-duality of finite centralizing extensions and a theorem of Haack-Fuller about Morita duality of semi-group rings.