In mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent x there exists an element y such that xy is a non-zero idempotent (Kaplansky 1968, pages 19, 25).
Kaplansky (1951) named them after Max August Zorn, who studied a similar condition in (Zorn 1941).
For associative rings, the definition of Zorn ring can be restated as follows: the Jacobson radical J(R) is a nil ideal and every right ideal of R which is not contained in J(R) contains a nonzero idempotent.
Replacing "right ideal" with "left ideal" yields an equivalent definition.
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