Well-Defined Maps on Equivalence Classes of Rings Let $R$ be a unitary commutative ring such that $1 \neq 0$ and $S \subseteq R$ be closed under multiplication (i.e. $\forall x,y \epsilon S, xy \epsilon S$) and contain 1. We define the relation $E$ on $R$ x $S$ by $(a,s)E(b,t)$ if and only if there exists $x \epsilon S$ such that $xat = xbs.$ 


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*Let $R_S$ be the set of $E$-classes. If $(a, s)$ $\epsilon$ $D$ x $S$, we denote by $\bar{(a, s)}$ $\epsilon$ $R_S$ the $E$-class of $(a, s)$. Show that the map $(\bar{(a,s)},\bar{(b,t)}\to \bar{(ab,st)}$ is well-defined.

*Show that the map $(\bar{(a,s)},\bar{(b,t)}\to \bar{(at+bs,st)}$ is well-defined. 

I'm really confused as to how to prove a map is well-defined. I think it means that the map holds regardless of the choice of a, b, s, and t but how do you prove that? Any help would be appreciated.
Thanks!
 A: You have the right general idea about what it means. Let’s look at the first one. To show that the map 
$$\left\langle\overline{\langle a,s\rangle},\overline{\langle b,t\rangle}\right\rangle\mapsto\overline{\langle ab,st\rangle}$$
is well-defined, you must show that if $\langle a,s\rangle\mathrel{E}\langle a',s'\rangle$ and $\langle b,t\rangle\mathrel{E}\langle b',t'\rangle$, so that $\overline{\langle a,s\rangle}=\overline{\langle a',s'\rangle}$ and $\overline{\langle b,t\rangle}=\overline{\langle b',t'\rangle}$, then $\overline{\langle ab,st\rangle}=\overline{\langle a'b',s't'\rangle}$, i.e., $\langle ab,st\rangle\mathrel{E}\langle a'b',s't'\rangle$. In other words, we want to show that the operation yields the same result no matter which names for the arguments we use.
To do this, assume that $\langle a,s\rangle\mathrel{E}\langle a',s'\rangle$ and $\langle b,t\rangle\mathrel{E}\langle b',t'\rangle$; by definition this means that there are $x,y\in S$ such that $xas'=xa's$ and $ybt'=yb't$. We want to show that there is a $z\in S$ such that
$$z(ab)(s't')=z(a'b')(st)\;;\tag{1}$$
can you use $x$ and $y$ to find a $z\in S$ satisfying $(1)$?
You can use the same general approach for the second one.
