IJPAM: Volume 20, No. 4 (2005)

A NOTE ON JACOBSON RINGS
WHICH ARE HOPFIAN

Satya Prakash Tripathi$^1$, Peter Zvengrowski$^2$
$^1$Department of Mathematics
K.M. College
University of Delhi
Delhi, 110007, INDIA
e-mail: [email protected]
$^2$Department of Mathematics and Statistics
University of Calgary
2500 University Drive N.W., Calgary, Alberta, T2N1N4, CANADA
e-mail: [email protected]


Dedicated to Professor K. Varadarajan
on the occasion of his seventieth birthday.


Abstract.A ring $R$ with unity $1$ is called Jacobson if, for each $x\in R$, there exists an exponent $n_x>1$ such that $x^{n_x}=x$. It is called Hopfian if every surjective endomorphism of $R$ is an isomorphism. The first objective of this note is to prove that for any Hopfian-Jacobson ring $R$, the polynomial ring $R[T]$ is also Hopfian (and several related results), thereby generalizing a previous theorem of the first author. The second objective is the construction of uncountably many non-isomorphic Jacobson rings, of unbounded exponents $n_x$, which are Hopfian and also non-Noetherian. This not only furnishes many non-trivial examples to which the above theorem applies, but also substantially enlarges the family of known Jacobson rings.

Received: April 22, 2005

AMS Subject Classification: 16S34, 16P99, 16U99

Key Words and Phrases: Jacobson rings, Hopfian rings, Boolean rings, polynomial rings, Lucas sequences of the first kind

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 20
Issue: 4