SOME REMARKS ON THE FINITENESS CONDITIONS OF RINGS

Authors

  • AHMED A. M. KAMAL Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt.

DOI:

https://doi.org/10.5556/j.tkjm.22.1991.4594

Keywords:

regular left-self-injective associative ring, Finiteness conditions, directly finite idempotents, central idem- potents, injective cover, ring of polynomials, essential left ideal, abelian idempotents

Abstract

The aim of this paper is to study the finiteness of rings. We prove that if $A$ is a regular left-self-injective ring, then $A$ is of type III (purely infinite) implies that $E(A[x])$ is, and $A$ contains an abelian idempotent if and only if $E(A[x])$ contains an abelian idempotent. Also we prove that.

If $A$ is a regular left self-injective ring and $J$ is a left ideal in $A[x]$ such that $C(J)$ is an essential left ideal in $A$, then there exists a countably generated left ideal $J'$ in $A[x]$ such that $C(J')$ is an essential left ideal in $A$, and if $J'$ is an essential left ideal in $A[x]$, then $J$ is an essential left ideal in $A[x]$.

References

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Ahmed A. M. Kamal, "Regular left self-injective rings of type I," Afrika Matematika, Journal of the african mathematical union (to appear).

Ahmed A. M. Kamal, "Semiprirneness of polynomial rings" (to appear).

I. Kaplansky, Rings of operators (Benjamin, New York, 1968).

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Published

1991-06-01

How to Cite

KAMAL, A. A. M. (1991). SOME REMARKS ON THE FINITENESS CONDITIONS OF RINGS. Tamkang Journal of Mathematics, 22(2), 165-174. https://doi.org/10.5556/j.tkjm.22.1991.4594

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Section

Papers