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In this article, we get classification theorems for a Ricci soliton on thepseudo-Riemannian hypersurface of the Minkowski space $\mathbb{E}^4_1$ takingthe potential vector field as the tangent component of the position vector ofthe pseudo-Riemannian hypersurface, denoted by $(M,g,{\bf x}^T,\lambda)$ inboth Riemannian and Lorentzian settings. First, we obtain the necessary andsufficient condition that a pseudo-Riemannian hypersurface $(M,g)$ in$\mathbb{E}^4_1$ admits a Ricci soliton $(M,g,{\bf x}^T,\lambda)$. In each ofthe form of the shape operator of a pseudo-Riemannian hypersurface, we obtaincharacterization a Ricci soliton on a pseudo-Riemannian hypersurface. Moreprecisely, we show that totally umbilical hypersurfaces, hyperbolic and apseudo-spherical cylinder in $\mathbb{E}^4_1$ is a shrinking Ricci solitonwhose the potential vector field is the tangent part of the position vector.Furthermore, we conclude that there exists only a shrinking Ricci soliton on aLorentzian isoparametric hypersurface in $\mathbb{E}^4_1$ withnondiagonalizable shape operator whose the minimal polynomial has double realroots.

Ricci Solitons on Pseudo-Riemannian Hypersurfaces of 4-dimensional Minkowski space