Show that $R$ is a ring

Question

Let $R$ a set with $+$ and $\cdot $ s.t.

1) $(R,+)$ is a group (not necessarily comutative)

2) $\cdot $ is associative, and distributive for $+$, i.e. $a\cdot (b\cdot c)=(a\cdot b)\cdot c$, $a\cdot (b+c)=a\cdot b+a\cdot c$ and $(a+b)\cdot c=a\cdot c+b\cdot c$

3) there is $1\in R$ s.t. $1\cdot x=x\cdot 1=x$.

Show that $R$ is a ring.

I really don't know how to did it. I think that I have to show that $(R,+)$ is commutative, but I didn't success.

score 6 · Accepted Answer · answered Feb 28 '17 at 14:29

6

Hint:

Compute $$ (1 + 1) \cdot (a + b) $$ in two ways, using the two distributive laws.

answered Feb 28 '17 at 14:29

Andreas Caranti

68,873

we get $a+a+b+b=(a+b)+(a+b)$ but I don't get the trick... – user386627 Feb 28 '17 at 14:41
Now exploit the fact that $(G, +)$ is a group, so that you can simplify. In other words, add $-a$ *on the left* and $-b$ *on the right*. – Andreas Caranti Feb 28 '17 at 14:47
Very nice. thanks – user386627 Feb 28 '17 at 15:00

Show that $R$ is a ring

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