Rings all of whose finitely generated ideals are automorphism-invariant

Abstract

Rings in which each finitely generated right ideal is automorphism-invariant (right[Formula: see text]-rings) are shown to be isomorphic to a formal matrix ring. Among other results it is also shown that (i) if [Formula: see text] is a right nonsingular ring and [Formula: see text] is an integer, then [Formula: see text] is a right self injective regular ring if and only if the matrix ring [Formula: see text] is a right [Formula: see text]-ring, if and only if [Formula: see text] is a right automorphism-invariant ring and (ii) a right nonsingular ring [Formula: see text] is a right [Formula: see text]-ring if and only if [Formula: see text] 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 [Formula: see text] is a right nonsingular right [Formula: see text]-ring if and only if [Formula: see text] is a left nonsingular left [Formula: see text]-ring.