Preheat oven to 350 degrees F. Grease and flour 9 x 2-inch-round layer-cake pan.
Mix eggs with fork in large bowl until well blended. Mix in brown sugar. Add oil in steady stream, mixing constantly until mixture is thick and smooth. Stir in lemon zest, rum extract, vanilla, ginger and salt.
Add baking soda in pinches, breaking up any lumps with fingers; stir in thoroughly. Stir in flour just until blended. Stir in pineapple.
Pour batter into prepared pan.
Bake in preheated 350 degree F oven for 30 to 35 minutes or until wooden pick inserted in center comes out clean. Cool cake in pan on wire rack for 10 minutes. Invert onto rack to cool completely.
When cake is completely cooled, frost with Lemon Icing. Makes 8 servings.
In large bowl, cut butter into confectioners sugar with pastry blender until butter is size of small grains. Beat in lemon juice, milk and lemon extract. Beat until mixture is smooth and well blended.