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Classification of the FRW universe with a cosmological constant and a perfect fluid of the equation of state $p = wρ$

We systematically study the evolution of the Friedmann-Robertson-Walker (FRW) universe coupled with a cosmological constant $Λ$ and a perfect fluid that has the equation of state $p=wρ$, where $p$ and $ρ$ denote, respectively, the pressure and energy density of the fluid, and $w$ is an arbitrary real constant. Depending on the specific values of $w,\; Λ$, and the curvature $k$ of 3-dimensional space, we separate all of the solutions into various cases. In each case the main properties of the evolution are given in detail, including the periods of deceleration and/or acceleration, and the existence of big bang, big crunch, and big rip singularities. In some cases, errors in classification and interpretation appearing in standard textbooks have been corrected.