Verified answer

No. I assume your ring is commutative. Suppose R is a commutative ring with identity and suppose x is a zero divisor and a unit as well. Since x is a zero divisor we can find a nonzero element z such that xz =zx = 0.

On the other hand since x is a unit we can find v in R such that vx = xv = 1.

Multiplying the equation xz =0 by v yields: v(xz ) =0 so (vx)z = 0. But vx = 1 and thus z = 0. This contradicts the assumption that z was nonzero.

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