Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-02T01:57:06.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Idempotent Ideals in Perfect Rings

Published online by Cambridge University Press:  20 November 2018

Gerhard Michler
Gerhard Michler*
Affiliation:
Universität Tubingen, Tubingen, West Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

All rings considered in this note have an identity element, and all R-modules are unitary.

Bass (2) defined a left perfect ring as a ring R satisfying the minimum condition on principal right ideals. A commutative ring R is perfect if and only if R is a direct sum of finitely many local rings whose radicals are T-nilpotent. Therefore, the commutative perfect rings with finite global projective dimension are just the direct sums of finitely many commutative fields, and hence they trivially satisfy the minimum condition for all ideals. However, in the non-commutative case, even hereditary perfect rings are not necessarily right or left artinian (cf. Example 3.4).

Each left perfect ring R has only finitely many idempotent (two-sided) ideals (Corollary 2.3), where the ideal X of R is called idempotent, if X = X2. Hence, it makes sense to consider minimal idempotent idealsof the left perfect ring R, i.e., ideals of R which are minimal in the set of all idemponent ideals of R.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 301 - 309
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Azumaya, G., On maximally central algebras, Nagoya Math. J. 2 (1951), 119–150.Google Scholar
2. Bass, H., Finistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.Google Scholar
3. Chase, S. U., A generalization of the ring of triangular matrices, Nagoya Math. J. 18 (1961), 13–25.Google Scholar
4. Harada, M., Note on dimension of modules and algebras, J. Inst. Poly tech. Osaka City Univ. Ser. A 7 (1956), 17–27.Google Scholar
5. Herstein, I., Theory of rings, Univ. Chicago Math. Lecture Notes, Chicago, Illinois, 1961.Google Scholar
6. Jacobson, N., Structure of rings, Amer. Math. Soc. Colloq. Publ., Vol. 37 (Amer. Math. Soc, Providence, R.I., 1956).Google Scholar
7. Kaplansky, I., Projective modules, Ann. of Math. (2) 68 (1958), 372–377.Google Scholar
8. Silver, L., Non-commutative localizations and applications, J. Algebra 7 (1967), 44–76.Google Scholar