WhenXis a compact subset of Cnand 0 <α≤ 1, we define subalgebras of the Lipschitz algebras Lip(X, α) andlip(X, α), which are generated by polynomials, rational functions with poles offX, functions which are analytic in some neighbourhood ofX, or functions which are analytic in the interior ofX. We investigate their maximal ideal spaces and the equality among them. For certain compact plane setsX, we extend a result due to Dales and Davie, by showing that every continuously differentiable function onXcan be approximated by rational functions inlip(X, α) norm. Finally we show that the algebra of rational functions with poles offXis dense inlip(X, α) on any compact plane setX, with planar measure zero, and then extend this result for particular compact subsets of Cn,n> 1.