With reference to Exercise 12, suppose we drop the condition that T have no divisors of zero and just require that nonempty T not containing 0 be closed under multiplication. The attempt to enlarge R to a commutative ring with unity in which every nonzero element of T is a unit must fail if T contains an element a that is a divisor of 0, for a divisor of 0 cannot also be a unit. Try to discover where a construction parallel to that in the text but starting with R x T first runs into trouble. In particular, for R = Z₆ and T = { 1, 2, 4}, illustrate the first difficulty encountered.

Data from Exercise 12

Let R be a nonzero commutative ring, and let T be a nonempty subset of R closed under multiplication and containing neither 0 nor divisors of 0. Starting with R x T and otherwise exactly following the construction in this section, we can show that the ring R can be enlarged to a partial ring of quotients Q(R, T). Think about this for 15 minutes or so; look back over the construction and see why things still work.