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In this paper we introduce an algebra embedding $\iota:K< X >\to S$ from thefree associative algebra $K< X >$ generated by a finite or countable set $X$into the skew monoid ring $S = P * \Sigma$ defined by the commutativepolynomial ring $P = K[X\times N^*]$ and by the monoid $\Sigma = < \sigma >$generated by a suitable endomorphism $\sigma:P\to P$. If $P = K[X]$ is any ringof polynomials in a countable set of commuting variables, we present also ageneral Gr\"obner bases theory for graded two-sided ideals of the gradedalgebra $S = \bigoplus_i S_i$ with $S_i = P \sigma^i$ and $\sigma:P \to P$ anabstract endomorphism satisfying compatibility conditions with ordering anddivisibility of the monomials of $P$. Moreover, using a suitable grading forthe algebra $P$ compatible with the action of $\Sigma$, we obtain a bijectivecorrespondence, preserving Gr\"obner bases, between graded $\Sigma$-invariantideals of $P$ and a class of graded two-sided ideals of $S$. By means of theembedding $\iota$ this results in the unification, in the graded case, of theGr\"obner bases theories for commutative and non-commutative polynomial rings.Finally, since the ring of ordinary difference polynomials $P = K[X\times N]$fits the proposed theory one obtains that, with respect to a suitable grading,the Gr\"obner bases of finitely generated graded ordinary difference ideals canbe computed also in the operators ring $S$ and in a finite number of steps upto some fixed degree.

Skew polynomial rings, Groebner bases and the letterplace embedding of the free associative algebra