Is there anything (even something weird or fancy) that you can multiply by zero and not get zero?

Question

I'm wondering if there's any kind of "imaginary anti-grassmann" (for lack of a better idea) or some strange object or other in math that you can multiply by zero and somehow not get something other than just zero?

For clarity, the answer I'm looking for would be something like: "0m = m Where 'm' is a Foo-number, which is a special object that keeps it's state when multiplied by zero, and here's how it interacts with normal numbers."

I'm looking to see if someone has invented such an object and built an algebra for it, etc.

Answer 1

This is not exactly what you want but I think it is worth to note (and is too long to be a comment).

By convention we have $$\large\color{red}{0^0=1}$$

Somehow exponentiation (by integers) is a kind of multiplication and here $0$ multiplied $0$ times by himself gives $1$. I have to say that the first time I encountered this, it was a very weird feeling and is probably the same as the one I would have by seeing an example which would answers positively your question. Anyway this stays a convention and other answers showed that the axioms we are currently using in mathematics don't leave space for such aliens. Moreover, I'm not sure how interesting would be a theory which allows the existence of this kind of objects.

Answer 2

The name "zero" is usually reserved for in every setting to be the additive identity.

I.e. that $x+0=0+x=x$

In any multiplicative ring we have the following then:

$$\begin{align} x\cdot 0 &= x\cdot 0 & \text{by reflexivity}\\ x\cdot (0+0) & =x\cdot 0 & \text{by additive identity}\\ x\cdot 0 + x\cdot 0 & = x\cdot 0 &\text{by distributivity}\\ x\cdot 0 & =0 & \text{by cancellation} \end{align}$$

Answer 3

Distributivity of multiplication over addition gives you the fact that the additive identity (zero) multiplied by any other element is $0$: $$0x=\left(0+0\right)x=0x+0x=0$$ See Example of "ring" without the distributive property? or near-ring.