What kind of structure is this: finite sequences with concatenation and multiplication. Let $(M, \times_M)$ be a monoid, call $M^*$ the finite sequences of elements of $M$.  For two sequences $u$, $v$, and an element $m \in M$, define:


*

*$u+v$: the sequence consisting of the elements of $u$ then the elements of $v$

*$m\times u$ is the sequence the $i$-th element of which is $m\times_M u_i$, with $u_i$ the $i$-th element of $u$

*Similarly for $u \times m$.


Hence $+$ has a neutral element (the empty sequence), and $\times$ has a neutral element (the neutral element of $M$) but no absorbing element in general.
Question:  What kind of structure is $(M^*, +, \times)$?
 A: The problem is that your operation $\times$ is not defined on $M^*$. One way to define it is to set
\begin{align}
(u_1 + \dotsm + u_r)(v_1 + \dotsm + v_s) &= (u_1 + \dotsm + u_r) \times v_1 + \dotsm + (u_1 + \dotsm + u_r) \times v_s \\
&= u_1v_1 + \dotsm + u_rv_1 + \dotsm + u_1v_s + \dotsm + u_rv_s
\end{align}
The structure $(M^*, +, *)$ could now be called a near-semiring (with unit) since it satisfies the following conditions:


*

*$(M^*, +)$ is a (noncommutative) monoid. Its identity is the empty sequence, which should be denoted by $0$ in this additive notation.

*$(M^*, \times)$ is a monoid with identity $1$.

*Multiplication distributes on the left over addition: for all $x,y,z ∈ M^*$, $z(x + y) = zx + zy$. 


Note that multiplication does not distribute on the right! 
Partially related bibliography
[1] B. Banaschewski and E. Nelson, On the non-existence of injective near-ring modules, Canad. Math. Bull. 20,1 (1977), 17–23.
[2] A. Fröhlich, On groups over a d.g. near-ring. I. Sum constructions and free R-groups, Quart. J. Math. Oxford Ser. (2) 11 (1960), 193–210.
[3] A. Fröhlich, On groups over a d.g. near-ring. II. Categories and functors, Quart. J. Math. Oxford Ser. (2) 11 (1960), 211–228.
[4] J.-É. Pin, Newton's forward difference equation for functions from words to words, CiE 2015, LNCS 9136 (2015), 71-82
A: I don't know if there is a more succinct way to characterize this structure, but I would say that it is:


*

*The free monoid over the set $M$: $M^*$

*On which the monoid $M$ acts left and right by multiplication.

