Let $(R,m,k)$ be a complete, reduced, local ring of dimension $1$ with infinite $k$. Let $I$ be a $m$-primary ideal. A Cohen-Macaulay module $M$ is called $I$-Ulrich if any of the following equivalent conditions is satisfied:

1. $M\cong IM$.
2. $IM=xM$ for any minimal reduction of $I$.
3. $e_I(M) = \ell(M/IM)$.
4. $M$ is a module over the blow-up ring $B(I): =\cup I^n:_{Q(R)}I^n$.

Problem: study the consequence of existence of $I$-Ulrich modules on both $M$, $I$ more deeply.