TITLE: What are zero divisors used for?
QUESTION [22 upvotes]: This is the first time I hear this term. Specifically the assertion is that $\mathbb{Z}$ has no zero divisors. So, from my understanding this is because there are not two non-zero numbers $a,b\in \mathbb{Z}$ such that $ab=0$. 
Also I can see that this definition is related to the one I learnt in high school that $a$ divides $0$ if $\exists b\in\mathbb{Z}\ (ba=0)$, the difference being that we need to consider the restriction $a,b\neq 0$ when dealing with zero divisors.
What is the motivation of zero divisors? what are they used for?

REPLY [7 votes]: Just to add something new to what is already said in the previous answers: in some cases you have "useful" zero-divisors. For example, let $V$ be a vector space and let $W$ be a subspace. Consider $A=\mathrm{End}(V)$, the ring of linear endomorphisms of $V$, endowed with the usual pointwise sum and the composition product. You can define a nontrivial endomorphism $\pi : V \to V$ in $A$ such that $\pi(V) = W$ and $\pi_{|W} = 1_W$. This is what you call a "projection", and it is idempotent, that is $\pi \circ \pi = \pi$. It is also a zerodivisor: if it weren't, then you would have $\pi \circ(1-\pi)=0$ and then $\pi = 0$ or $\pi = 1_V$. It should be pointed out that the existence of nontrivial idempotents on a ring is useful to obtain decomposition results for modules over the given ring. Read, for example, this article:
http://en.wikipedia.org/wiki/Idempotent_element#Role_in_decompositions