Checking commutativity of a diagram of modules over some ring and what the commutativity of the diagram implies.

Question

Suppose that you have the following diagram of modules over some ring:

These are my questions:

(1) To prove that the diagram is commutative, we needs to prove that $gf=kh$, $wf=rv$, $zh=uv$, $sw=xg$, $xk=tz$, am I right?

(2) Suppose that the above diagram is commutative. Do the following equalities hold: $sr=tu$, $swf=xkh$

The multiplication above means composition of maps

Thanks in advance.

Answer 1

$sr=tu$ needs to be included in the conditions, it does not follow automatically. To see this let all the other maps be 0.

$swf=xkh$ follows from $sw=xg$ and $gf=kh.$

Answer 2

What if $E,G,H$ all equal $\Bbb Z$ with $u,t$ the identity map, and all other rings are the zero ring. I think that might be a counter-example.