Coding for the synchronous noiseless T-user real adder channel is cons
idered by employing cyclic codes with symbols from an arbitrary finite
integer ring. The code construction is based on the factorisation of
x(n) - 1 over the unit ring of an appropriate extension of a finite in
teger ring. Any number of users in the system can be independently act
ive and the maximum achievable sum rate is 1 (when all T users are act
ive).