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Let $\mathbf{k}$ be a field of arbitrary characteristic, let $\Lambda$ be afinite dimensional $\mathbf{k}$-algebra, and let $V$ be an indecomposablefinitely generated non-projective Gorenstein-projective left $\Lambda$-modulewhose stable endomorphism ring is isomorphic to $\mathbf{k}$. In this article,we prove that the universal deformation rings $R(\Lambda,V)$ and$R(\Lambda,\Omega_\Lambda V)$ are isomorphic, where $\Omega_\Lambda V$ denotesthe first syzygy of $V$ as a left $\Lambda$-module. We also prove the followingresult. Assume that $\Lambda$ is Gorenstein and that $\Gamma$ is anotherGorenstein $\mathbf{k}$-algebra such that there exists $\ell \geq 0$ and a pairof bimodules $({_\Gamma}X_\Lambda, {_\Lambda}Y_\Gamma)$ that induces a singularequivalence of Morita type with level $\ell$ (as introduced by Z. Wang) between$\Lambda$ and $\Gamma$. Then the left $\Gamma$-module $X\otimes_\Lambda V$ isalso Gorenstein-projective with stable endomorphism ring isomorphic to$\mathbf{k}$ and the universal deformation ring $R(\Gamma, X\otimes_\Lambda V)$is isomorphic to $R(\Lambda, V)$.

Singular equivalences of Morita type with level, Gorenstein algebras, and universal deformation rings