Vol. 30, No. 3, 1969

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
Zero square rings

Richard Peter Stanley

Vol. 30 (1969), No. 3, 811–824
Abstract

A ring R for which x2 = 0 for all x R is called a zero-square ring. Zero-square rings are easily seen to be locally nilpotent. This leads to two problems: (1) constructing finitely generated zero-square rings with large index of nilpotence, and (2) investigating the structure of finitely generated zerosquare rings with given index of nilpotence. For the first problem we construct a class of zero-square rings, called free zero-square rings, whose index of nilpotence can be arbitrarily large. We show that every zero-square ring whose generators have (additive) orders dividing the orders of the generators of some free zero-square ring is a homomorphic image of the free ring. For the second problem, we assume Rn0 and obtain conditions on the additive group R+ of R (and thus also on the order of R). When n = 2, we completely characterize R+. When n > 3 we obtain the smallest possible number of generators of R+, and the smallest number of generators of order 2 in a minimal set of generators. We also determine the possible orders of R.

Mathematical Subject Classification
Primary: 16.32
Milestones
Received: 9 September 1968
Published: 1 September 1969
Authors
Richard Peter Stanley