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Learning more abstract algebra, really not the most enjoyable of subjects, as nothing seems all that clear cut, but here goes anyway.

I have a set $\mathbb Q = \{{p \over q} : p,q\in \mathbb Z \text{ and } q \neq 0\}$ which is the set of rational numbers and for $x,y\in \mathbb Q$ defined the binary operation $*$ on $\mathbb Q$ by $$x*y = x + y + xy.$$

The pair $(\mathbb Q, * )$ has an identity element. Find the identity element then verify the identity properties for $(\mathbb Q,*)$.

Where do I begin?

Ludolila
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TheCrownedPixel
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  • You are looking for a rational number y so that in your multiplication formula, everything except for x disappears(so that multiplying by y didn't do anything to x!). – QTHalfTau May 29 '15 at 18:53
  • See also http://math.stackexchange.com/questions/980196/show-that-a-star-b-a-cdot-bab-is-binary-operation-for-the-group-bbb-q. – Dietrich Burde May 29 '15 at 19:02

2 Answers2

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The identity element $e$ is defined by $x*e=x$ and $e*y=y$ for all $x,y$. Therefore solve the equation

$$x+e+xe=x$$

You get

$$e+xe=0$$ $$e(x+1)=0$$

For this to be true for all $x$,

$$e=0$$

Doing it the other way,

$$e+y+ey=y$$ $$e+ey=0$$ $$e(1+y)=0$$ $$e=0$$

You "verify the identity properties" by showing that it is indeed true that $x*e=x$ and $e*y=y$ for all $x,y$. I'll leave that to you.

Rory Daulton
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Let $y$ be the identity element.

Then $x*y=x+y+xy=x$ for all $x \in Q$.

Now, let $x=0$. What do you conclude?

Matthew Conroy
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