  A ring $A$ is called Hopfian if $A$ cannot be isomorphic to a proper
homomorphic image $A/J$. $A$ is called Bassian if there cannot be an injection
of $A$ into a proper homomorphic image $A/J$. We consider classes of Hopfian
and Bassian rings, and tie representability of algebras and chain conditions on
ideals to these properties. In particular, any semiprime algebra satisfying the
ACC on semiprime ideals is Hopfian, and any semiprime affine PI-algebra over a
field is Bassian.
