Determine the total ring of fractions Determine the total ring of fractions of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}_{12}$.
 A: If $R$ is a commutative ring, then $Q(R)$, the total ring of fractions of $R$, is defined as being the ring of fractions $S^{-1}R$, where $S$ is the set of regular elements (or non-zerodivisors) of $R$. 
In general, for the ring $\mathbb Z_n$, $n\ge 2$, the regular elements coincide with the invertible elements, and thus $Q(\mathbb Z_n)=\mathbb Z_n$.
For $R=\mathbb Z\times \mathbb Z$ the set of regular elements is $S=(\mathbb Z\setminus\{0\})\times(\mathbb Z\setminus\{0\})$, and then $Q(R)=S^{-1}R\simeq\mathbb Q\times\mathbb Q$.
Remark. A generalization of the above result is the following: If $R=R_1\times\cdots\times R_n$, where $R_i$ are integral domains with fraction fields $Q(R_i)=K_i,$ then $Q(R)=K_1\times\cdots\times K_n$.
A: Hints:


*

*Rephrase your question so it isn't an order.

*Determine the units and zero-divisors of both rings. Remember that the only units of $\Bbb Z$ are $\pm1$ and that the units of $\Bbb Z_n$ are the representatives coprime to $n$. Anything that is a not a zero-divisor which is not a unit will get turned into a unit in the total ring of fractions.

*One of these rings is already its own total ring of fractions.
