The Feedback with Carry Shift Registers (FCSRs) have been proposed as an alternative to Linear Feedback Shift Registers (LFSRs) for the design of stream ciphers. FCSRs have good statistical proper- ties and they provide a built-in non-linearity. However, two attacks have shown that the current representations of FCSRs can introduce weak- nesses in the cipher. We propose a new "ring" representation of FCSRs based upon matrix definition which generalizes the Galois and Fibonacci representations. Our approach preserves the statistical properties and circumvents the weaknesses of the Fibonacci and Galois representations. Moreover, the ring representation leads to automata with a quicker diffu- sion characteristic and better implementation results. As an application, we describe a new version of F-FCSR stream ciphers.