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In book R. Göbel, P. Hill, Wolfgang "Abelian Group Theory and Related Topics", I found next Beaumont -Pierce Principal theorem: Any torsion-free ring $R$ of finite rank is quasi-equal to $S\oplus N$, where $N$ is nilradical, and $S$ is subring of $R$.

My question is: is $R$ need to be commutative in this theorem. If "No", what nilradical mean for noncimmutative rings?

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No, every endomorphism ring of a torsion-free group falls into this class of rings and they are usually non-commutative.

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