Question

Let R be a ring with identity and $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{R}$ . Assume that neither a nor b is a zero divisor. If $\mathit{a}\mathit{b}$is a unit, prove that $\mathit{a}$and$\mathit{b}$ are units. [Hint: Exercise 21.]

Short answer

It is proved that $a$and $b$are units.

Step-by-step solution

Statement of Exercise 37

Exercise 37 considers R as a ring with identity and $a,b\in R$. Suppose that a is not a zero divisor. Show that $ab=1_R$if and only if$ba=1_R$ .

Show that aand bare units

Assume that $c\in R$in which $\left(ab\right)c=1_R=c\left(ab\right)$.

It is observed that $a\left(bc\right)=1_R$from associativity.

Moreover, bc is not a zero divisor because when $\left(bc\right)d=0$then,

$$\begin{gathered} 1_{R}d=abcd \\ =a{0}_R \\ =0_R \end{gathered}$$

This means that 1 is a zero divisor, which would be a contradiction.

As a result,$\left(bc\right)a=1_R$ from exercise 37. Hence, a is a unit.

Likewise, it is obtained $\left(ca\right)b=1_R$that from associativity, which indicates that ca is not a zero divisor. As a result,$b\left(ca\right)=1_R$ . Hence, b is a unit.

Hence, it is proved that a and b are units.