Rings in which nilpotents form a subring

Šter, Janez

Abstract

carpathian_2016_32_2_251_258_abstract

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carpathian_2016_32_2_251_258

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Let $R$ be a ring with the set of nilpotents $\Nil(R)$ . We prove that the following are equivalent:
(i) $\Nil(R)$ is additively closed,
(ii) $\Nil(R)$ is multiplicatively closed and $R$ satisfies K\”othe’s conjecture,
(iii) $\Nil(R)$ is closed under the operation $x\krogec y=x+y-xy$ ,
(iv) $\Nil(R)$ is a subring of $R$ .
Some applications and examples of rings with this property are given, with
an emphasis on certain classes of exchange and clean rings.