Rings all of whose finitely generated ideals are automorphism-invariant

Abstract

Rings in which each finitely generated right ideal is automorphism-invariant (rightfa-rings) are shown to be isomorphic to a formal matrix ring. Among other results it is also shown that (i) if R is a right nonsingular ring and n > 1 is an integer, then R is a right self injective regular ring if and only if the matrix ring Mn(R) is a right fa-ring, if and only if Mn(R) is a right automorphism-invariant ring and (ii) a right nonsingular ring R is a right fa-ring if and only if R is a direct sum of a square-full von Neumann regular right self-injective ring and a strongly regular ring containing all invertible elements of its right maximal ring of fractions. In particular, we show that a right semiartinian (or left semiartinian) ring R is a right nonsingular right fa-ring if and only if R is a left nonsingular left fa-ring.