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Let $\KX =K\langle X_1,\ldots ,X_n\rangle$ be the free algebra generated by$X=\{ X_1,\ldots ,X_n\}$ over a field $K$. It is shown that with respect to anyweighted $\mathbb{N}$-gradation attached to $\KX$, minimal homogeneousgenerating sets for finitely generated graded (two-sided) ideals of $\KX$ canbe algorithmically computed, and that if an ungraded (two-sided) ideal $I$ of$\KX$ has a finite Gr\"obner basis $\G$ with respect to a graded monomialordering on $\KX$, then a minimal standard basis for $I$ can be computed viacomputing a minimal homogeneous generating set of the associated graded ideal$\langle\LH (I)\rangle$.

Computation of Minimal Homogeneous Generating Sets and Minimal Standard Bases for Ideals of Free Algebras