Question

Let $a$be a nonzero element of a ring$R$ with identity. If the equation$ax=1_R$ has a solution$u$ and the equation$ya=1_R$ has a solution$v$ , prove that u=v .

Short answer

It isproved that $u=v$.

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

The given ring with identity is$R$ with the non-zero element$a$ .

Let the elements be$a,bandc$ in the given ring, such that $a\ne 0_R$.

Also, it is given that: $au=1_R$and$va=1_R$

Now, let us evaluate as:

$\begin{array}{rcl}va& =& \frac{1_R}{u}\\ \left(va\right)u& =& 1_{R}u\\ v\left(au\right)& =& u\\ v{1}_R& =& u\\ v& =& u\end{array}$

Hence it is proved that$u=v$ .