On the arithmetic of the endomorphisms ring End(Zp×Zp2)

Abstract

For a prime number p, Bergman (Israel J Math 18:257–277, 1974) established that End(Zp×Zp2) is a semilocal ring with p 5 elements that cannot be embedded in matrices over any commutative ring. We identify the elements of End(Zp×Zp2) with elements in a new set, denoted by E p , of matrices of size 2 × 2, whose elements in the first row belong to Zp and the elements in the second row belong to Zp2 ; also, using the arithmetic in Zp and Zp2 , we introduce the arithmetic in that ring and prove that the ring End(Zp×Zp2) is isomorphic to the ring E p . Finally, we present a Diffie-Hellman key interchange protocol using some polynomial functions over E p defined by polynomials in Z[X].