Requirements for a*b = 0 implying a = 0 or b = 0

[pyrotix]

I can think of an proof using greater thans or less thans for an ordered field. For complex numbers I believe I know the outlines proof using properties of magnitude.

Obvious the statement isn't true for matrices. What at minimum do you need for the statement to be true? I'm thinking you may need a metric with certain properties of transformation when the * is applied.

But I'm an engineering student really, so this isn't my area of expertise.

 

[Hurkyl]

What at minimum do you need for the statement to be true?

The axiom

a*b=0 implies a=0 or b=0​

A trivial answer, but nonetheless the exact answer to the question you asked.

I suppose what you really mean is just to query what other sorts of properties imply this property. In ring theory, a ring with this property is called a "domain", or sometimes "integral domain". I can think of relationships of the property to the notion of a prime ideal, but I suspect that wouldn't be too useful to you.

As a special case, ring theory the complex numbers are a domain because:

• The real numbers are a domain
• The polynomial $x^2 + 1$ doesn't factor over the real numbers

And for the same reason ($x^2-1$ does factor over the real numbers), the split complex numbers are not a domain.

And because every complex polynomial factors into linears, there are no algebraic extensions of the complex numbers which are (commutative) domains. e.g. we cannot create a new ring with division by adding in a new number q that satisfies some algebraic identity, as we did to construct the complexes from the reals.

 

[micromass]

Also note that the statement

$$a*b=0~\Rightarrow~a=0~\text{or}~b=0$$

is equivalent to the cancellation property: (in commutative setting at least)

$$ac=bc~\Rightarrow~a=b$$

Furthermore, every field is an integral domain. In less high-brow terminology: if every nonzero element a has an inverse a^-1 (for which of course holds aa^-1=a¹a=1[), then the statements above hold.

This is the reason why

$$a*b=0~\Rightarrow~a=0~\text{or}~b=0$$

holds in the real, complex and rational numbers: because every nonzero element has an inverse. In the matrices, not every element has an inverse, thus the above does not hold.

Note that not only fields satisfy

$$a*b=0~\Rightarrow~a=0~\text{or}~b=0$$

Consider the integers for example...