Structure of T-rings. (English) Zbl 0588.16021
Radical theory, Proc. 1st Conf., Eger/Hung. 1982, Colloq. Math. Soc. János Bolyai 38, 633-655 (1985).
[For the entire collection see Zbl 0575.00009.]
The authors call an associative ring R with identity a (left) T-ring if \(Ext_ R(A,B)\neq 0\) whenever A is a non-projective left R-module and B is a non-injective one. In the case, if R is not completely reducible, there exists just one non-projective simple module N and T-rings can be divided into three types: (1) \(Soc_ N(R)\neq 0\); (2) \(Soc_ N(R)=0\) and Soc(R) is a left direct summand of R; (3) \(Soc_ N(R)=0\) and Soc(R) is not a left direct summand of R. Besides some basic properties of T-rings, the authors give a full description of T-rings of type (3).
The authors call an associative ring R with identity a (left) T-ring if \(Ext_ R(A,B)\neq 0\) whenever A is a non-projective left R-module and B is a non-injective one. In the case, if R is not completely reducible, there exists just one non-projective simple module N and T-rings can be divided into three types: (1) \(Soc_ N(R)\neq 0\); (2) \(Soc_ N(R)=0\) and Soc(R) is a left direct summand of R; (3) \(Soc_ N(R)=0\) and Soc(R) is not a left direct summand of R. Besides some basic properties of T-rings, the authors give a full description of T-rings of type (3).
Reviewer: L.Bican
MSC:
| 16Exx | Homological methods in associative algebras |
| 16S20 | Centralizing and normalizing extensions |
| 16D80 | Other classes of modules and ideals in associative algebras |
