Question 1.
( a ) ⇒ ( b ) {\displaystyle (a)\Rightarrow (b)~} is fine.
but ( b ) ⇒ ( a ) {\displaystyle (b)\Rightarrow (a)~} should be modified to:
if R n → f R → 0 {\displaystyle R^{n}\xrightarrow {f} R\to 0~} is an exact sequence then f {\displaystyle f~} is given by a unimodular row.
The simple example that makes statement of ( b ) ⇒ ( a ) {\displaystyle (b)\Rightarrow (a)~} false is :
0 → Z → × 2 Z {\displaystyle 0\to \mathbb {Z} \xrightarrow {\times 2} \mathbb {Z} ~}
is fine.
should be modified to:
is an exact sequence then 
is given by a unimodular row.
false is :
