# Proof: let $A$ a ring, then $(-a) \cdot (-b) = a \cdot b$ $\forall a,b \in A$

I must prove this property:

Property: let $A$ a be ring, then $(-a) \cdot (-b) = a \cdot b$, $\forall a,b \in A$.

Proof: let $a \in A$ and $b \in A$, by hypothesis $A$ is a ring then $a \cdot 0=0$ and $b + (-b) = (-b) + b= 0$ and $(-b) \in A$, therefore $a \cdot (b + (-b))=0$, but by hypothesis $\cdot$ is distributive then $a \cdot (b + (-b))=a \cdot b + a \cdot (-b) =0$, therefore $-(a \cdot (-b))=a \cdot b$, but in a ring is true that $-(c \cdot d)=(-c) \cdot d$, $\forall c,d \in A$, therefore we have $(-a) \cdot (-b)=-(a \cdot (-b))=a \cdot b$.

It is correct?

In light of MathGem's answer, let's look at your statement again. You said, "in a ring it is true $-(a\cdot b)=(-a)\cdot b$." What you're saying is that $$\tag 1 a\cdot b+a\cdot (-b)=0$$

which is true.

Thus, you in turn claim that in a ring

$$a\cdot b+(-a)\cdot b=0$$

which is true. Then, it follows

$$\tag 2 (-a)\cdot (-b)+a\cdot (-b)=0$$

But since inverses are unique $(1),(2)$ imply $a\cdot b=(-a)\cdot(-b)$.

There is a nice proof, that goes as follows:

Look at the expression $$a\cdot b+a\cdot (-b)+(-a)\cdot(-b)$$ and use the distributive laws in two different ways. One would be the above is$$a\cdot b+a\cdot (-b)+(-a)\cdot(-b)= \\a\cdot (b+(-b))+(-a)\cdot(-b)=\\a\cdot 0+(-a)\cdot (-b)=\\ 0+(-a)\cdot (-b)=\\(-a)\cdot (-b)$$

What is the other?

• okok but my process is correct?? Your proof is interesting!!! Thanks soo much!!
– mle
Apr 15, 2013 at 0:59
• @GarnakOlegovitc Sorry, yes. Your proof looks OK.
– Pedro
Apr 15, 2013 at 0:59
• $a \cdot b + a \cdot (-b) + (-a) \cdot (-b) = a \cdot b + (a + (-a)) \cdot (-b) = a \cdot b + 0 \cdot (-b) = a \cdot b$ Correct?
– mle
Apr 15, 2013 at 1:04
• @GarnakOlegovitc Correct. All in all, $(-a)\cdot(-b)=a\cdot b$.
– Pedro
Apr 15, 2013 at 1:04
• I don't think Math Gems is Bill, Peter. The registration date on the account predates Bill's account being suspended. Although their 'styles' are quite similar... Apr 15, 2013 at 2:06

There is an error. The statement "but in a ring is true..." begs the principle. Instead, let $\rm\:b = -c\:$ in what you just proved. More explicitly, you have proved $\rm\:x(-\color{#c00}y) = -(xy)\:$ for all $\rm\:x,y.\:$ Thus, applying this twice we deduce $\rm\ (-a)(-\color{#c00}b) = -((-a)b) = -(b(-\color{#c00}a)) = -(-(ba)) = ba = ab.$

A more conceptual way to view the proof is that $\rm\:(-a)(-b)\:$ and $\rm\:ab\:$ are both inverses of $\rm\:(-a)b,\:$ hence they are equal by uniqueness of inverses.

Equivalently, evaluate $\ \rm\overline{(-a)(-b)\! \,+\,} \overline{ \underline {(-a)b}} \underline{\,+\,ab_{\phantom{_{1}}}}\:$in two ways, over or underlined first.

• Why do $a,b$ get swapped in your first paragraph?
– Pedro
Apr 15, 2013 at 1:59
• After writing out the OPs statement, and hoping I interpret it alright, it doesn't look wrong.
– Pedro
Apr 15, 2013 at 2:06
• @Peter So one may apply the law just proved, which pulls the negation out of the second argument of a product. Alternatively, one could extend the law by using commutativity, i.e. extend it to $\rm\: −(xy)=x(−y)=(−y)x,\:$ so one can pull the negation out of either argument. Apr 15, 2013 at 2:15
• @Peter Without any proof of the claim "but in a ring it is true that P(c,d)" the proof is either incomplete or circular. That the OP thinks this statement does not require proof indicates a possible conceptual error, since that statement is equivalent to the statement he just proved. Apr 15, 2013 at 2:17
• True. But as I see it, he observes $a\cdot b+(-a)\cdot b=0$ and $c\cdot d+c\cdot (-d)=0$, then lets $c=-a$, $d=-b$.
– Pedro
Apr 15, 2013 at 2:19