# Ring without zero divisors

*From*: Ben <benv@xxxxxxxx>*Date*: 1 May 2007 12:07:22 -0700

Hi,

I have following question.

Is it true, that if we have a ring R (with unit) without zero divisors

satisfying that

(*) for all x there is y such that yx^3y = x,

then R is divison ring?

I can write these observations

- Two sided ideals of R are idempotent (I^2=I)

- The elements y and x^4 (as above) commute, because x^4y = xx^3y =

yx^3yx^3y = yx^3 x = yx^4.

Thank you for your ideas how to prove this or for your counterexamples!

.

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