Question

If $u$is a unit in a ring$R$ with identity, prove that $u$is not a zero divisor.

Short answer

It is proved that the given unit in a ring$R$with identity is not a zero divisor.

Step-by-step solution

Elements of Rings 

If a ringhas an element $k$, such that $k\in R$.

Then the following equation will have a unique solutionas:

$k+x=0_R$

Proof using contradiction:

The given ring with identity is$R$ , and its unit is$u$ .

Let the unit$u$be the zero divisor in the given ring.

And $u^{-1}$ be the multiplicative inverse of $u$, where $u\ne 0_R$.

Now, there exists an element $x$such that:

$\begin{array}{rcl}\frac{u^{-1}}{ux}& =& 0_R\left\{x\ne 0_R\right\}\\ u^{-1}ux& =& u^{-1}{0}_R\\ 1_{R}x& =& 0_R\\ x& =& 0_R\\ & & \end{array}$.

But, $x\ne 0_R$.

This leads to a contradiction.

Hence it is proved that the given unit in a ring $R$ with identity is not a zero divisor.