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 × T first runs into trouble. In particular, for R = ℤ6 and T = {1. 2, 4}, illustrate the first difficulty encountered. [Hint: It is in Step 1 .]
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