Understanding the definition of a distraction of a monomial ideal Let $I$ be a monomial ideal in the ring $\mathbb{C}[\partial]=\mathbb{C}[\partial_1,...,\partial_n]$, a commutative polynomial subalgebra of the Weyl algebra $D_n$ of linear partial differential operators on $\mathbb{C}[x_1,...,x_n]$ where $\partial_i$ denotes the operator $\partial/\partial x_i$. Let $\theta_i=x_i\partial_i$, so that $\mathbb{C}[\theta]=\mathbb{C}[\theta_1,...,\theta_n]$ is also a commutative polynomial subalgebra of $D_n$. The distraction of $I$ is defined as $$\tilde{I}= (\mathbb{C}(x)\otimes_{\mathbb{C}[x]}D\cdot I)\cap \mathbb{C}[\theta]\subseteq \mathbb{C}[\theta]$$
In short, I have no clue what $(\mathbb{C}(x)\otimes_{\mathbb{C}[x]}D\cdot I)$ is trying to say. I'm guessing that $\otimes$ is some kind of generalization of tensor products to algebras (I also need an explanation of how this generalization works). I don't know what $\otimes_{\mathbb{C}[x]}$ means. Also, I'm guessing that $D\cdot I=\{di : d\in D \wedge i\in I\}$.
EDIT
Something else to clarify is that $\mathbb{C}(x)$ denotes the field of complex numbers. Also, my intuition about the meaning of $D\cdot I$ was essentially correct. However, the accepted answer defines $D\cdot I$ in a way that shows the significance of such a construction.
 A: *

*$D \cdot I$ : this is the $D$-module generated by $I$. Its elements are sums of the form $d_1i_1 + d_2i_2 + \cdots + d_ki_k$, where $d_j \in D$ and $i_j \in I$ for all $j$. 

*$D \cdot I$ is a module over $D_n$ and thus over $\mathbb C[x]$. The tensor product here is of modules. The tensor product formula given reduces to
$$\mathbb{C}(x) \otimes_{\mathbb C[x]} D\cdot I = \mathbb{C}(x_1,\ldots, x_n) \cdot (D \cdot I) \subseteq \mathbb{C}(x_1,\ldots,x_n) \langle \partial_1,\ldots, \partial_n\rangle,$$
where the right-hand side is polynomial differential operators with rational (edit: i.e. rational function) coefficients.
If $M$ is a right $A$-module and $N$ is a left $A$-module, then the tensor product $M \otimes_A N$ is the abelian group generated by symbols $m \otimes n$ for $m \in M$ and $n \in N$, modulo the additivity relations $(m_1 + m_2)\otimes n \sim m_1 \otimes n + m_2\otimes n$ and similarly on the right, as well as $m \otimes an \sim ma \otimes n$ for $a \in A, m\in M, n \in N$. If $A$ is commutative, then $M \otimes_A N$ will be an $A$-module. 
The multiplication map $\mathbb{C}(x) \times D \cdot I \to \mathbb{C}(x_1,\ldots, x_n)\langle \partial_1,\ldots,\partial_n\rangle$ is bilinear, so it induces the desired map in the highlighted equation above. This map is an isomorphism (leading me to write an equals sign above) since it can be checked the image has the desired universal property.
