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. Left or right Artinian rings , left or right perfect rings , semiprimary rings and von Neumann regular rings are all examples of associative Zorn rings.
References
Kaplansky, Irving (1951), "Semi-simple alternative rings" , Portugaliae Mathematica , 10 (1): 37– 50, MR 0041835 Kaplansky, I. (1968), Rings of Operators Tuganbaev, A. A. (2002), "Semiregular, weakly regular, and π-regular rings", J. Math. Sci. (New York) , 109 (3): 1509– 1588, doi :10.1023/A:1013929008743 , MR 1871186 , S2CID 189870092 Zorn, Max (1941), "Alternative rings and related questions I: existence of the radical", Annals of Mathematics , Second Series, 42 (3): 676– 686, doi :10.2307/1969256 , JSTOR 1969256 , MR 0005098 