Geodesic
On two-sided artinian quotient rings
Glasgow mathematical journal, Tome 13 (1972) no. 2, pp. 159-163

Voir la notice de l'article provenant de la source Cambridge University Press

Djabali [1] has proved that, if R is a right and left noetherian ring with an identity and if the proper prime ideals of R are maximal, then R has a right and left artinian two-sided quotient ring. Robson [5, Theorem 2.11] and Small [6, Theorem 2.13] have proved independently that, if Ris a commutative noetherian ring, then Rhas an artinian quotient ring if and only if the prime ideals of Rthat belong to the zero ideal are all minimal. We shall generalise these results by proving theTheorem. Let R be a right and left noetherian ring with a regular element. Then R has a right and left artinian two-sided quotient ring if and only if each prime ideal of R consisting of zero–divisors is minimal. 
Smith, P. F. On two-sided artinian quotient rings. Glasgow mathematical journal, Tome 13 (1972) no. 2, pp. 159-163. doi: 10.1017/S0017089500001592
@article{10_1017_S0017089500001592,
     author = {Smith, P. F.},
     title = {On two-sided artinian quotient rings},
     journal = {Glasgow mathematical journal},
     pages = {159--163},
     year = {1972},
     volume = {13},
     number = {2},
     doi = {10.1017/S0017089500001592},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500001592/}
}
TY  - JOUR
AU  - Smith, P. F.
TI  - On two-sided artinian quotient rings
JO  - Glasgow mathematical journal
PY  - 1972
SP  - 159
EP  - 163
VL  - 13
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500001592/
DO  - 10.1017/S0017089500001592
ID  - 10_1017_S0017089500001592
ER  - 
%0 Journal Article
%A Smith, P. F.
%T On two-sided artinian quotient rings
%J Glasgow mathematical journal
%D 1972
%P 159-163
%V 13
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500001592/
%R 10.1017/S0017089500001592
%F 10_1017_S0017089500001592

[1] 1.Djabali, M., Anneaux de fractions généralisés artiniens, C. R. Acad. Sci. Paris 268 (1969), 2138–2140. Google Scholar

[2] 2.Goldie, A. W., Rings with maximum condition, Mimeographed lecture notes, Yale University, 1961. Google Scholar

[3] 3.Goldie, A. W., Lectures on non-commutative noetherian rings, Canad. Math. Congress, York University, Toronto, 1967. Google Scholar

[4] 4.Jacobson, N., The theory of rings, Amer. Math. Soc. Surveys No. 2 (New York, 1943). Google Scholar | DOI

[5] 5.Robson, J. C., Artinian quotient rings, Proc. London Math. Soc. (3) 17 (1967), 600–616. Google Scholar | DOI

[6] 6.Small, L. W., Orders in artinian rings, J. Algebra 4 (1966), 13–41. Google Scholar | DOI

[7] 7.Small, L. W., On some questions in noetherian rings, Bull. Amer. Math. Soc. 72 (1966), 853–857. Google Scholar | DOI

Cité par Sources :