On the Structure of Neat-injective Envelopes
by
Arshad Imam
Dokuz Eylul Uni., Izmir, Turkey
Coauthors: Refail Alizade, Karen D. Akinci

It is well-known that over any ring R any module has an injective envelope. Injective envelopes of abelian groups and their structure are described in Chapter 24, [2]. Existence of neat-injective envelopes is proved by Onishi [3]. We give a description of the neat-injective envelope of an abelian group A in terms of the basic subgroup Bp of the p-component Tp(A).

Firstly we describe the neat-injective envelope of the group Z_pⁿ(n > 1) with prime p. Since basic subgroups Bp of Tp(A) is a direct sum of cyclic groups , we then described the neat-injective envelopes of the basic group Bp.

In the next theorem we give the structure of neat-injective envelopes of any p-groups in terms of Bp and Ap/Bp. Since every torsion group is a direct sum of its p-component, we find out neat-injective envelope for any torsion group. After that we give the stucture of neat-injective envelopes for torsion-free groups. Finally with the help of neat-injective envelopes of torsion and torsion-free group we determine the neat-injective envelope for any group A.

REFERENCES

[1] D.K. Harrison, J.M. Irwin, C.L. Peercy and E.A. Walker.(1963), High extension of abelian group Acta Math. Acad. Sci. Hunger. 14, 319-330.

[2] Fuchs L. (1970) Infinite Abelian Groups Vol 1. New York and London, Academic Press.

[3] Onishi, M. On minimal neat-injective groups containing a given group as a neat subgroup. 33(1984), Comment Math . Univ. St.Paul. No.2, 203-207.

Date received: May 30, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafe-08.