Hereditary Artinian rings of finite representation type
Hereditary Artinian rings of finite representation type
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有限表示类型的遗传性 Artinian 环
DOI：
10.1007/bfb0088466
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发表时间：
1980
影响因子：
1.3
通讯作者：
D. Simson
中科院分区：
文献类型：
--
作者：
P. Dowbor;C. Ringel;D. Simson
R e c l l [1 - 1,6] t ha t h e R e d我t y R f i n t e d实施爱奥那岛l lgebra is f我n i t e i型和o n y l i f R公司荷兰国际集团(ing)反应图我s d s j o n t联合国离子Dynkin图的n, n, n c, d n, e 6, 7, 8 e, f 4 G 2 occour ing在l ie t h e o R y。在这里，我们把它看成是一种基因，一种基因，一种基因，一种基因，一种基因，一种基因，一种基因，一种基因，一种基因，一种基因，一种基因，一种基因，一种基因。我t t u rns t ha t I s f我n t e I型和l y我f C (A)我s d s j o n t联盟Coxeter图n, n (= Cn), d n e 6 e 7 ES, F4, G2, H3, H4a I2 ~ p), C l s s我f y r r e d u C I B l e Coxeter组[3]。然而,e x i s t e n c e r n g s H型3 H 4我2 (p) w i t H p = 5 o r p ~ 7仍然开放:i t取决于r t H e r d我f f c u l t问题我ons担忧ing d v s i o n r i n g s。另一方面，我们可以把它用在任意一个Coxeter图的“分支系统”上，这个分支系统是Oynkin图的基础系统。维类型的一个h e r e d i t y一个r t我n n r ng f我n t e r e p r e s e n t t i o n型j u s t形成这样一个分支系统。P.Dowbor和o . simson在他们的论文[9,101]和1979年渥太华会议上宣布了其中的1 / 2和1 / 5是由P.Dowbor和o . simson发现的。1978年，英国皇家科学院（C.M.R inge）在伦敦举行的世界学术会议上宣布了这一消息，并于1978年在伦敦宣布了这一消息。
R e c a l l [ 1 1 , 6 ] t ha t a h e r e d i t a r y f i n i t e d imens iona l a lgebra i s of f i n i t e type i f and o n l y i f a co r respond ing diagram i s a d i s j o i n t un ion of the Dynkin diagrams A n , B n, C n, D n, E 6, E 7, E 8, F 4, G 2 occour ing in L ie t h e o r y . Here, we w i l l c o n s i d e r the gene ra l case of a h e r e d i t a r y a r t i n i a n r i ng A and assoc ia te to i t a diagram F(A) . I t t u rns out t ha t A i s of f i n i t e type i f and on l y i f C(A) i s the d i s j o i n t union of the Coxeter diagrams A n , B n ( = Cn), D n, E 6, E 7, ES, F4, G2, H3, H4a I2~ p) which c l a s s i f y the i r r e d u c i b l e Coxeter groups [ 3 ] . However, the e x i s t e n c e of r i n g s of type H 3, H 4 and I 2 ( p ) w i t h p = 5 o r p~ 7 remains open: i t depends on r a t h e r d i f f i c u l t ques t i ons concern ing d i v i s i o n r i n g s . On the o t h e r hand, we d e f i n e f o r any Coxeter diagram "branch system" which g e n e r a l i z e the roo t systems of the Oynkin d iagrams. The dimension types of a h e r e d i t a r y a r t i n i a n r i ng of f i n i t e r e p r e s e n t a t i o n type j u s t form such a branch system. The r e s u l t s of s e c t i o n s 1, 2 and 5 were ob ta ined by P.Dowbor and O.Simson who announced pa r t of them in the papers ~9, 101] and at the Ottawa Conference 1979. The r e s u l t s were ob ta ined i n d e p e n d e n t l y by C.M.R inge l who announced them at the Oberwol fach meet ing on d i v i s i o n r i n g s in 1978.