Equations in Algebras
  We show that the Diophantine problem(decidability of equations) is undecidable in free associative algebras over any field and in the group algebras over any field of a wide variety of torsion free groups, including toral relatively hyperbolic groups, right angled Artin groups, commutative transitive groups, the fundamental groups of various graph groups, etc. 
Olga Kharlampovich, Alexei Myasnikov
math.LO