No need to peel the ginger--it will be strained out of the mixture before serving.
Bring first 5 ingredients to a boil, stirring until sugar dissolves. Reduce heat, and simmer 8 minutes. Remove from heat; let stand 1 hour.
Pour pineapple juice mixture through a wire-mesh strainer into a large pitcher, discarding ginger. Stir in lime rind, and, if desired, rum. Serve over crushed ice.