Question

Let Rbe a ring with identity. If ab and a are units in R, prove that b is a unit.

Short answer

It is proved that b is a unit

Step-by-step solution

Definition of units

An element,$a$ in aring,Rhas an identity known as theunitwhen there exists $u\in R$in which $au=1_R=ua$. In this case, the element $u$is the inverseof a and is represented by $a^{-1}$.

Show that b is a unit

Consider that $a,b\in R$with the ring $R$has an identity and a (multiplicative inverse:$a^{-1}$), (multiplicative inverse: c) are units. Then,

$$\begin{gathered} b\left(ca\right)=1_{R}bca \\ =a^{-1}abca \\ =a^{-1}\left(ab\right)ca \\ =a^{-1}\left(\left(ab\right)c\right)a \\ =\left(a^{-1}{1}_R\right)a \\ =a^{-1}a \\ =1_R \end{gathered}$$

To prove that $\left(ca\right)b=1_R$as follows

$$\begin{gathered} \left(ca\right)b=\left(c{1}_R\right)\left(ab\right) \\ =c\left(a{a}^{-1}\right)ab \\ =ca\left(a^{-1}a\right)b \\ =ca{1}_{R}b \\ =c\left(a{1}_R\right)b \\ =cab \\ =c\left(ab\right) \\ =1_R \end{gathered}$$

It is observed that ca is a multiplicative inverse of b, That is, b is a unit.

Hence, it is proved that b is a unit.