Vol. 20, No. 3, 1967

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Vol. 286: 1  2
Vol. 285: 1  2
Vol. 284: 1  2
Vol. 283: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Subscriptions
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Endomorphism rings of primary abelian groups

Robert William Stringall

Vol. 20 (1967), No. 3, 535–557
Abstract

This paper is concerned with the study of certain homomorphic images of the endomorphism rings of primary abelian groups. Let E(G) denote the endomorphism ring of the abelian p-group G, and define H(G) = {α E(G)|x G,px = 0 and height x < imply height α(x) > height x}. Then H(G) is a two sided ideal in E(G) which always contains the Jacobson radical. It is known that the factor ring E(G)∕H(G) is naturally isomorphic to a subring R of a direct product Πn=1Mn with n=1Mn contained in R and where each Mn is the ring of all linear transformations of a Zp space whose dimension is equal to the n 1 Ulm invarient of G. The main result of this paper provides a partial answer to the unsolved question of which rings R can be realized as E(G)∕H(G).

Theorem. Let R be a countable subring of Π0Zp which contains the identity and 0Zp. Then there exists a p. group G with a standard basic subgroup and containing no elements of infinite height such that E(G)∕H(G) is isomorphic to R. Moreover, G can be chosen without proper isomorphic subgroups; in this case, H(G) is the Jacobson radical of E(G).

Mathematical Subject Classification
Primary: 20.30
Milestones
Received: 20 January 1966
Published: 1 March 1967
Authors
Robert William Stringall