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On subrings of matrix rings over fields

Published online by Cambridge University Press:  24 October 2008

J. -E. Björk
J. -E. Björk
Affiliation:
University of Stockholm, Sweden
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Extract

Let K be a commutative field and let V be an n-dimensional vector space over K. We denote by L(V) the ring of all K-linear endomorphisms of V into itself. A subring of L(V) is always assumed to contain the unit element of L(V), but it need not be a vector subspace of the K-algebra L(V). Suppose now that A is a subring of L(V). Then we may consider L(V) as a left or a right A-module. We may also consider V as a left A-module, i.e. if x ∈ V and if f ∈ A then we get the image element f(x) in V. If we choose a basis for V over K then we get the associated matrix representation of L(V). So in this way L(V) is identified with the full matrix ring Mn(K), and in this way the study of subrings of L(V) can be reduced to the study of subrings of Mn(K).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 2 , September 1970 , pp. 275 - 284
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Artin, E., Nesbitt, C. and Thrall, R.Rings with minimum condition (University of Michigan, 1944).Google Scholar
(2)Bass, H.Finistic homological dimension and a homological generalization of semiprimary rings. Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
(3)Björk, J.-E.Rings satisfying a minimum condition on principal ideals. J. Reins Angew. Math. 236 (1969), 112–119.Google Scholar
(4)Björk, J.-E.Rings satisfying certain chain conditions. J. Reins Angew. Math. (in the Press).Google Scholar
(5)Björk, J.-E. Conditions which imply that subrings of artinian rings are artinian (to appear).Google Scholar
(6)Jacobson, N.Structure of rings. Amer. Math. Soc. Coll. Publ. 37 (1956).Google Scholar
(7)Osofsky, B. L.A generalization of Quasi-Frobenius rings. J. Algebra 4 (1966), 373–388.CrossRefGoogle Scholar