Abstract: An infinite-dimensional -graded k-algebra A is called projectively simple if for every nonzero two-sided ideal . We show that if a projectively simple ring A is strongly noetherian, is generated in degree 1, and has a point module, then A is equal in large degree to a twisted homogeneous coordinate ring . Here X is a smooth projective variety, is an automorphism of X with no proper -invariant subvariety (we call such automorphisms wild), and is a -ample line bundle. We conjecture that if X admits a wild automorphism then every irreducible component of X is an abelian variety. We prove several results in support of this conjecture; in particular, we show that the conjecture is true if . In the case where X is an abelian variety, we describe all wild automorphisms of X . Finally, we show that if A is projectively simple and admits a balanced dualizing complex, then is Cohen–Macaulay and Gorenstein. [Copyright &y& Elsevier]