
Rings and zero divisors
Let R be a commutative ring with a 1≠ 0, and let S denote the set of nonzerodivisors of R (that is {s∈R  sr≠ 0 for all r∈R\0}).
Prove that S is a multiplicatively closed subset of R which contains 1 but not 0.
Prove that every element of S1R is either a unit (that is, u such that uv = 1 for some v) or a zero divisor (zero divisor includes 0)
So far I got that subset S which is multiplicatively 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 of all non zero divisors, but if it isn’t, then I imagine S1R will contain elements which are neither units nor zero divisors. But I'm not sure how to show this... any help??