Zorn ring

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, New York: W. A. Benjamin, Inc.
• 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