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Let $R$ be a (commutative or non-commutative) unital ring, $X$ be a non-empty set, and $R \langle\! \langle X \rangle\! \rangle$ be the ordered series ring (in fact, a ring of formal power series over $R$ in $|X|$ non-commuting variables) obtained by endowing the set of all functions $\mathscr F(X) \to R$ with the usual operations of pointwise addition and Cauchy product. Here, $\mathscr F(X)$ is the free monoid on $X$, whose operation (that is, word concatenation) I'll denote by $\ast$.

While looking for a counterexample to a certain property in the class of local rings, I happened to note (with the help of Daniel Smertnig, who pointed out a problem with my previous formula) that, for each $z \in X$, the mapping $\partial_z$ that sends an ordered series $f \in R \langle\! \langle X \rangle\! \rangle$ to the function $$ \mathscr F(X) \to R \colon \mathfrak z \mapsto f(\mathfrak z) {\sum}_{(\mathfrak u, \mathfrak v) \in \mathscr F(X) \times \mathscr F(X): \mathfrak u \ast z \ast \mathfrak v = \mathfrak z} \delta_{u \ast v}, $$ is a well-defined derivation of $R \langle\! \langle X \rangle\! \rangle$ (here, $\delta_{u \ast v}$ means the Kronecker delta centered at the $X$-word $u \ast v$). In particular, the Leibniz identity follows from the fact that $\mathscr F(X)$ is a cancellative monoid with trivial group of units and every $X$-word factors uniquely in $\mathscr F(X)$ as a product of elements of $X$ (that is, $X$-words of length one).

My question is whether anyone here can offer a reference where $\partial_z$ is being introduced: I thought I would have found $\partial_z$ defined in Cohn's book on FIRs (where ordered series rings are discussed in Sect. 1.5), but it's not there (as far as I can see). I've also tried with Lam's A First Course in Noncommutative Rings, but the conclusion is the same.

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    $\begingroup$ Did you look at Berstel-Reutenauer's book on rational power series? This kind of derivation or variants thereof are used in formal language theory to study factorizations and I would probably guess Schutzenberger used it somewhere. It is very closely related to the way he shows how to recognize products is regular languages using Schutzenberger products but it connects more directly with marked products. You might contact Reutenauer. He would know. $\endgroup$ Mar 1 at 12:33
  • $\begingroup$ @BenjaminSteinberg Thanks for the pointers. Actually, I had already tried with Berstel and Reutenauer's Noncommutative rational series and applications (I was addressed to the same book by Daniel Smertnig, though for a different reason), and there doesn't seem to be much about derivations in it. In any case, I'll try to write to Reutenauer. $\endgroup$ Mar 1 at 19:01
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Following Benjamin Steinberg's advice (in the comments under the OP), I contacted Christophe Reutenauer, who pointed out that the derivation $\partial_z$ in the OP is, of course, well known and its first explicit occurrence in the literature dates back at least to the work of Claude Lenormand. In particular, Reutenauer offered the following reference:

C. Lenormand, Exponentiation de la dérivation, et intégration des séries en variables non commutatives,Séminaire Schützenberge 1 (1969-70), Exposé No. 20, 7 pp. (freely available from EuDML),

where the set of variables is finite and the integers embed in the coefficient ring (not that this makes any real difference for the definition of $\partial_z$).

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