ON THE AUTOMORPHISM-INVARIANCE OF FINITELY GENERATED IDEALS AND FORMAL MATRIX RINGS
Abstract
In this paper, we study rings having the property that every
finitely generated right ideal is automorphism-invariant. Such rings are called
right fa-rings. It is shown that a right fa-ring with finite Goldie dimension is
a direct sum of a semisimple artinian ring and a basic semiperfect ring. From
this, we obtain that if R is a right fa-ring with finite Goldie dimension such
that every minimal right ideal is a right annihilator and the right it’s socle is
essential in RR, R is also indecomposable (as ring), not simple with non-trivial
idempotents then R is QF. In this case, QF-rings are the same as q-, fq-,
a-, fa-rings. We also obtain a result of the automorphism-invariance of formal
matrix rings.
finitely generated right ideal is automorphism-invariant. Such rings are called
right fa-rings. It is shown that a right fa-ring with finite Goldie dimension is
a direct sum of a semisimple artinian ring and a basic semiperfect ring. From
this, we obtain that if R is a right fa-ring with finite Goldie dimension such
that every minimal right ideal is a right annihilator and the right it’s socle is
essential in RR, R is also indecomposable (as ring), not simple with non-trivial
idempotents then R is QF. In this case, QF-rings are the same as q-, fq-,
a-, fa-rings. We also obtain a result of the automorphism-invariance of formal
matrix rings.
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