# Prove that for all a,b in a ring R, (-a)b=-ab=a(-b)

• Jul 6th 2011, 07:57 AM
dwsmith
Prove that for all a,b in a ring R, (-a)b=-ab=a(-b)
Prove that for all a,b in a ring R, \$\displaystyle (-a)b=-ab=a(-b)\$
The hint says use: \$\displaystyle a+(-a)=0\$

Since any integer has a negative, we know \$\displaystyle a+(-a)=0\$

Multiple by b:

\$\displaystyle (a+(-a))b=0(b)\Rightarrow ab+(-a)b=0\$

How can I show \$\displaystyle (-a)b=-ab=a(-b)\$ from the above?
• Jul 6th 2011, 09:05 AM
Chris L T521
Re: Prove that for all a,b in a ring R, (-a)b=-ab=a(-b)
Quote:

Originally Posted by dwsmith
Prove that for all a,b in a ring R, \$\displaystyle (-a)b=-ab=a(-b)\$
The hint says use: \$\displaystyle a+(-a)=0\$

Since any integer has a negative, we know \$\displaystyle a+(-a)=0\$

Multiple by b:

\$\displaystyle (a+(-a))b=0(b)\Rightarrow ab+(-a)b=0\$

How can I show \$\displaystyle (-a)b=-ab=a(-b)\$ from the above?

Interesting approach (I don't have a problem with it). Note that \$\displaystyle -ab=-(ab)\$. So the first part follows right away: \$\displaystyle (-a)b=-ab\$.

For the second part, you do pretty much the same thing; however, you start with \$\displaystyle b+(-b)=0\$ and then multiply by \$\displaystyle a\$ on the left:

\$\displaystyle a(b+(-b))=0\implies\ldots\$

Then because '=' is transitive, it now follows that \$\displaystyle (-a)b=-ab=a(-b)\$.

I hope this makes sense.
• Jul 6th 2011, 09:26 AM
Isomorphism
Re: Prove that for all a,b in a ring R, (-a)b=-ab=a(-b)
Quote:

Originally Posted by dwsmith
Prove that for all a,b in a ring R, \$\displaystyle (-a)b=-ab=a(-b)\$
The hint says use: \$\displaystyle a+(-a)=0\$

Since any integer has a negative, we know \$\displaystyle a+(-a)=0\$

integer?
I think you mean the ring is a group under addition and thus every element in the ring has an additive inverse.

Quote:

Multiple by b:

\$\displaystyle (a+(-a))b=0(b)\Rightarrow ab+(-a)b=0\$
Have you proved that \$\displaystyle b.0 = 0\$ for all elements b in the ring?

Quote:

How can I show \$\displaystyle (-a)b=-ab=a(-b)\$ from the above?