The graded version of Goldie’s theorem. (English) Zbl 0985.16031
Huynh, D. V. (ed.) et al., Algebra and its applications. Proceedings of the international conference, Athens, OH, USA, March 25-28, 1999. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 259, 237-240 (2000).
It is well known that Goldie’s theorems are of great importance for the study of left or right Noetherian rings. A first study of Goldie’s theorems for graded rings was done by I. D. Ion and C. Năstăsescu [Rev. Roum. Math. Pures Appl. 23, 573-588 (1978; Zbl 0382.16001)], and later in the book by C. Năstăsescu and F. Van Oystaeyen [Graded ring theory, North-Holland Math. Library, Vol. 28 (1982; Zbl 0494.16001)].
In the paper under review, the authors give a more general version for Goldie’s theorem (only the gr-prime case). More precisely, let \(R\) be a \(G\)-graded ring, where \(G\) is an Abelian (semigroup) group. If \(R\) is a gr-prime and left gr-Goldie ring then \(R\) has a gr-simple, gr-Artinian left ring of fractions. The proof of this theorem is very nice and elegant.
For the entire collection see [Zbl 0947.00022].
In the paper under review, the authors give a more general version for Goldie’s theorem (only the gr-prime case). More precisely, let \(R\) be a \(G\)-graded ring, where \(G\) is an Abelian (semigroup) group. If \(R\) is a gr-prime and left gr-Goldie ring then \(R\) has a gr-simple, gr-Artinian left ring of fractions. The proof of this theorem is very nice and elegant.
For the entire collection see [Zbl 0947.00022].
Reviewer: C.Năstăsescu (Bucureşti)
MSC:
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 16N60 | Prime and semiprime associative rings |
| 16P70 | Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras) |
