Rings with the contraction property
Abstract: A ring (not necessarily commutative or with unit) has the contraction property iff every ideal of every subring of is a contracted ideal.
It is shown that is a primitive ring with the contraction property iff 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 is a torsion free nil ring with the contraction property then . 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 is a nil ring with the contraction property then is torsion as an additive group.
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Keywords: Contracted ideal, nil ring, absolutely algebraic field
Article copyright: © Copyright 1971 American Mathematical Society