Let and let

Let $A= \mathbb{Q}\times \mathbb{Q}$ and let $\ast$ be a binary operation on A defined by $\left ( a,b \right )\ast \left ( c,d \right )= \left ( ac,b+ad \right )$ for $\left ( a,b \right ),\left ( c,d \right )\, \epsilon \, A\cdot$ Determine, whether $\ast$ is commutative and associative. Then, with respect to $\ast$ on A
(i) Find the identity element in A.
(ii) Find the invertible elements of A.

Answers (1)

given $A= \mathbb{Q}\times \mathbb{Q}$ and defined by $\left ( a,b \right )\ast \left ( c,d \right )= \left ( ac,b+ad \right )$ for $\left ( a,b \right ),\left ( c,d \right )\, \epsilon \, A\cdot$
Commutative: $\left ( a,b \right )\ast \left ( c,d \right )= \left ( ac,b+ad \right )$ = LHS
                       $\left ( c,d \right )\ast \left ( a,b \right )= \left ( ca,d+cb \right )= RHS$
$\because LHS\neq RHS$ Not commutative
Associative: $\left ( a,b \right )\ast \left [ \left ( c,d \right )\ast \left ( e,f \right ) \right ]$
        $= \left ( a,b \right )\ast \left ( ce,d+cf \right )$
      $\Rightarrow \left [ ace,b+ad+acf \right ]= LHS$
Now $\left [ \left ( a,b \right )\ast \left ( c,d \right )\ast \left ( e,f \right )= \left [ ac,b+ad \right ]\ast \left ( e,f \right ) \right ]$
$= \left [ ace,b+ad+acf \right ]= RHS$
$\because LHS= RHS\; \; \therefore$ It is associative
(i) $\left ( a,b \right )\ast e= \left ( a,b \right )$
$\Rightarrow a= ac \Rightarrow c= 1\; and\; b= b+ad\Rightarrow ad= 0$
$\Rightarrow d= 0$
$\therefore \left ( a,b \right )\ast \left ( 1,0 \right )= \left ( a,b+a\times 0 \right )= \left ( a,b \right )$
$\Rightarrow \left ( 1,0 \right )$ is the identity element.
$\left ( a,b \right )\ast \left ( c,d \right )= e= \left ( 1,0 \right )$
$\Rightarrow ac= 1\; and\; b+ad= 0$
$c= \frac{1}{a} \; \; \; d= \frac{-b}{a}$
Inverse $= \left ( \frac{1}{a},\frac{-b}{a} \right )\; \because D\neq 0\; \; Q\neq 0$
$\therefore$ all elements are invertible which have the following properly $\left ( a,b \right ),a\neq 0$