  We study the $\mathfrak{gl}_{1|1}$ supersymmetric XXX spin chains. We give an
explicit description of the algebra of Hamiltonians acting on any cyclic tensor
products of polynomial evaluation $\mathfrak{gl}_{1|1}$ Yangian modules. It
follows that there exists a bijection between common eigenvectors (up to
proportionality) of the algebra of Hamiltonians and monic divisors of an
explicit polynomial written in terms of the Drinfeld polynomials. In particular
our result implies that each common eigenspace of the algebra of Hamiltonians
has dimension one. We also give dimensions of the generalized eigenspaces. We
show that when the tensor product is irreducible, then all eigenvectors can be
constructed using Bethe ansatz. We express the transfer matrices associated to
symmetrizers and anti-symmetrizers of vector representations in terms of the
first transfer matrix and the center of the Yangian.
