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Rings with the contraction property


Author: William J. Wickless
Journal: Proc. Amer. Math. Soc. 27 (1971), 57-60
MSC: Primary 16.20
DOI: https://doi.org/10.1090/S0002-9939-1971-0266954-3
MathSciNet review: 0266954
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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.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0266954-3
Keywords: Contracted ideal, nil ring, absolutely algebraic field
Article copyright: © Copyright 1971 American Mathematical Society

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