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. References: 24
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