Yoshitomo Baba

In [10, Theorem 3.1] K. R. Fuller characterized indecomposable injective projective modules over artinian rings using i-pairs. In [3] the author generalized this theorem to indecomposable projective quasi-injective modules and indecomposable quasi-projective injective modules over artiniain rings. In [2] the author and K. Oshiro studied the above Fuller's theorem minutely. And M. Morimoto and T. Sumioka generzlized these results to modules in [17]. Further in [13] M. Hoshino and T, Sumioka extended the results in [3] to perfect rings and consider the condition "colocal pairs". Furthermore in [7] the anther studied the results in [3] from the point of view of [2] and [13] and gave results on colocal pairs. The purpose of this note is to report about this development.

We define Harada rings of a component type. The class of them contains indecomposable serial rings. We consider the structure of them and show that they have weakly symmetric self-dualities.

In [23] H. Tachikawa gave two theorems: Every maximal quotient QF-3 ring is represented as an endomorphism ring of some kind of module Every QF-3 ring is a special subring of the maximal quotient QF-3 ring. These theorems are a little sophisticated in [18]. On the other hand, in [3] Y. Baba and K. Iwase defined quasi-Harada rings (abbreviated QH rings) and showed that QH rings are QF-3. So now we have several rings which are also QF-3 rings: Serial rings, one-sided Harada rings, quasi-Harada rings, and (one-sided artinian) QF-2 rings. In this paper we apply the above two theorems for QF-3 rings to these rings. And we study the structure of these rings.

In [17, (4.3)Theorem] and [22, (9.4)Theorem] H. Tachikawa and C. M. Ringel gave a one-to-one correspondence between (I) Morita equivalence classes of semi-primary QF-3 maximal quotient rings R with l.gl.dimR less than or equal to 2, and (II) Morita equivalence classes of rings of finite representation type. And they showed that each semi-primary QF-3 maximal quotient ring R with I.gl.dimR less than or equal to 2 is characterized as the functor ring of a left, artinian ring of finite representation type. Further in [8, Theorem 4.2] K. R. Fuller and H. Hullinger showed that a ring A is serial iff its functor ring R is QF-2. Using the above Tachikawa and Ring el's result, we have a one-to-one correspondence between (I) Morita. equivalence classes of semi-primary QF-3 QF-2 maximal quotient rings R with l.gl.dimR less than or equal to 2, and (II) Morita equivalence classes of serial rings as the restriction of the above one-to-one correspondence. In this paper, we consider artinian rings R with l.gl.dimR less than or equal to 2. Main results are Theorems 3, 6 and 8. In Theorem 3 we gives an independent proof of the Fuller-Hullinger theorem and specify more precisely those rings which can occur as a functor ring of a serial ring, i.e., we show that the above one-to-one correspondence gave another one-to-one correspondence between (I) Morita equivalence classes of QH maximal quotient rings R with l.gl.dimR I 2, and (II) Morita equivalence classes of serial rings, hence semi-primary QF-3 QF-2 maximal quotient rings R with l.gl.dimR less than or equal to 2 are QH. Then combining with [6, Theorem 15] we obtain the structure of QH rings R with l.gl.dimR less than or equal to 2. In Theorem 6 we show that left or right H-rings R with l.gl.dimt2 less than or equal to 2 are serial rings. Last in Theorem 8 we describe those rings for which the functor ring is a serial ring.

M. Harada (''Ring Theory, Proceedings of 1978 Antwerp Conference,'' pp. 669-690, Dekker, New York, 1979) studied the following two conditions: (*) Every non-small left R-module contains a non-zero injective submodule. (*)* Every non-cosmall right R-module contains a non-zero projective direct summand. K. Oshiro (Hokkaido Math. J. 13, 1984, 310-338) further studied the above conditions, and called a left artinian ring with (*) a left Harada ring (abbreviated left H-ring) and a ring satisfying the ascending chain condition for right annihilator ideals with (*)* a right co-Harada ring (abbreviated right co-H-ring). K. Oshiro (Math. J. Okayamn Univ. 31, 1989, 161-178) showed that left PI-rings and right co-H-rings are the same rings. Here we are particularly interested in the following characterization of a left H-ring given in Harada's paper above: A ring R is a left I-I-ring if and only if R is a perfect ring and for any left non-small primitive idempotent e of R there exists a non-negative integer t(e) such that (a)(R)Re/S-k(Re-R) is injective for any k is an element of {0,..., t(e)} and (b) Re-R/S-te+1(Re-R) is a small module, where S-k(Re-R) denotes the k-th socle of the left R-module Re. This characterization implies (+) S-te+1(Re-R) is a uniserial left R-module for any left non-small primitive idempotent e in a left H-ring R. In this paper, we generalize left H-rings by removing (+). Concretely, since a left H-ring R is also characterized by the statement that R is left artinian and for any primitive idempotent g of R there exist a primitive idempotent e(g) of R and a non-negative integer k(g) such that the injective hull of the left R-modure Rg/Jg is isomorphic to Re/S-kp(Re-R(g)), where J is the Jacobson radical of (R)R, we define a more general class of rings by the condition that for any primitive idempotent g of R there exists a primitive idempotent e of R such that the injective hull of the left R-module Rg/Jg is isomorphic to Re/{x is an element of Re\gRx = 0}, and call it a left quasi-Harada ring (abbreviated left QH ring). In Section 1 we characterize a left QH ring by generalizing the characterizations of a left H-ring given by M. Harada and K. Oshiro. We also consider the weaker rings, left QF-2 rings and right QF-2* rings. K. Oshiro (Math. J. Okayama Univ. 32, 1990, 111-118) described the connection between left H-rings and QF rings. In Section 2 we describe the connection between two-sided QH rings and QF rings. (C) 1996 Academic Press, Inc.