The aim of this paper is on the death phenomena of coupled oscillators with very general coupling structures. The authors consider those couplings which possess the transferable property that the coupled information can be transformed directly or indirectly from any oscillator to any others. The death solution is obtained as a limit of upper solutions and lower solutions. They investigate a coupled cyclic chain of oscillators, in which the coupling is transferred in one direction and with a time lag. To obtain the asymptotic stability of the death solution, they establish necessary and sufficient conditions to ensure the zeros for a class of exponential polynomials to lie to the left of the imaginary axis.