Let $B$ be a commutative ring. Let $n$ be a strictly positive integer and $b _ { 1 } , \ldots , b _ { n }$ be elements of $B$. a) Show that there exists a unique ring morphism $f$ from $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$ to $B$ such that $f \left( X _ { i } \right) = b _ { i }$ for all $i \in \{ 1 , \ldots , n \}$. b) Deduce that $B$ has property (TF) if and only if there exist an integer $n \geq 1$ and a surjective ring morphism $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right] \rightarrow B$. c) Show that an abelian group $M$ has property (F) if and only if there exist an integer $r \geq 1$ and a surjective group morphism $\mathbf { Z } ^ { r } \rightarrow M$. d) Let $A$ and $B$ be commutative rings such that there exists a surjective ring morphism from $A$ to $B$. Show that if $A$ has property (TF), then so does $B$. State and prove an analogous statement for property (F).
Let $B$ be a commutative ring. Let $n$ be a strictly positive integer and $b _ { 1 } , \ldots , b _ { n }$ be elements of $B$.\\
a) Show that there exists a unique ring morphism $f$ from $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$ to $B$ such that $f \left( X _ { i } \right) = b _ { i }$ for all $i \in \{ 1 , \ldots , n \}$.\\
b) Deduce that $B$ has property (TF) if and only if there exist an integer $n \geq 1$ and a surjective ring morphism $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right] \rightarrow B$.\\
c) Show that an abelian group $M$ has property (F) if and only if there exist an integer $r \geq 1$ and a surjective group morphism $\mathbf { Z } ^ { r } \rightarrow M$.\\
d) Let $A$ and $B$ be commutative rings such that there exists a surjective ring morphism from $A$ to $B$. Show that if $A$ has property (TF), then so does $B$. State and prove an analogous statement for property (F).