I am currently trying to understand the MSE of ridge regression. First, I am calculating the MSE mathematically, but I found it quite vague. After reviewing some books and articles I understood that

$$
\begin{aligned}
\text{MSE}(\hat{\beta_R}) &= E[||\hat{\beta}_R-{\beta}||^2] \\
\Rightarrow\hat{\beta_{R}}-\beta &= ((X^TX+\lambda)^{-1}X^TX-I)\beta+e \\
\Rightarrow||\hat{\beta}_R-{\beta}||^2 &= (\hat{\beta_R}-{\beta})^T(\hat{\beta_R}-{\beta})
\end{aligned}
$$  

After that I got stuck because of the norm and expectation calculation. I tried to solve it, but it becomes so complicated.

I checked books like: "The Elements of Statistical Learning" and "An Introduction to Statistical Learning". 

Can anyone please clarify MSE of ridge regression or guide me to a good source?