Universal deformation rings of modules over Frobenius algebras

Let $k$ be a field, and let $\mathrm{\Lambda }$ be a finite dimensional $k$-algebra. We prove that if $\mathrm{\Lambda }$ is a self-injective algebra, then every finitely generated $\mathrm{\Lambda }$-module $V$ whose stable endomorphism ring is isomorphic to $k$ has a universal deformation ring $R(\mathrm{\Lambda },V)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. If $\mathrm{\Lambda }$ is also a Frobenius algebra, we show that $R(\mathrm{\Lambda },V)$ is stable under taking syzygies. We investigate a particular Frobenius algebra ${\mathrm{\Lambda }}_0$ of dihedral type, as introduced by Erdmann, and we determine $R({\mathrm{\Lambda }}_0,V)$ for every finitely generated ${\mathrm{\Lambda }}_0$-module $V$ whose stable endomorphism ring is isomorphic to $k$.