LetRbe a commutative ring with a 1≠ 0, and letSdenote the set of non-zerodivisors ofR(that is {s∈R|sr≠ 0 for allr∈R\0}).

Prove thatSis a multiplicative-ly closed subset ofRwhich contains 1 but not 0.

Prove that every element ofS-1Ris either a unit (that is,usuch thatuv= 1 for somev) or a zero divisor (zero divisor includes 0)

So far I got that subset S which is multiplicative-ly closed and does not contain zero divisors and obtain a larger ring in which the elements of S become units. I think we even get an appropriate universal property (with respect to the subset S of course). The last idea – that every element is either a unit or a zero divisor – I’m not so sure about. That might be true if S is the set ofallnon zero divisors, but if it isn’t, then I imagineS-1Rwill contain elements which are neither units nor zero divisors. But I'm not sure how to show this... any help??