Rings with the contraction property

Abstract:

A ring $R$ (not necessarily commutative or with unit) has the contraction property iff every ideal of every subring of $R$ is a contracted ideal. It is shown that $R$ is a primitive ring with the contraction property iff $R$ is an absolutely algebraic field. This result, together with the fact that the Jacobson Radical of a ring with the contraction property is nil, shows that a nil semisimple ring with the contraction property is a subdirect sum of absolutely algebraic fields (and is therefore commutative). It is shown that if $R$ is a torsion free nil ring with the contraction property then ${R^2} = (0)$. It follows that any torsion free ring with the contraction property is the extension of a zero ring and a subdirect sum of absolutely algebraic fields. Also, if $R$ is a nil ring with the contraction property then ${R^2}$ is torsion as an additive group.