Fulton 1.8

1.41.* If $S$ is module-finite over $R,$ then $S$ is ring finite over $R.$

Suppose $S$ is module finite over $R.$ Then $S=\{r_1s_1+\ldots+r_ns_n \ | \ r_i\in R\}$ for some $s_1,\ldots,s_n\in S.$ Thus $s=R[s_1,\ldots,s_n].$

1.42. Show $S=R[X]$ is ring-finite over $R,$ but not module finite.

Clearly $R[X]$ is ring finite by definition. Observe for any finite set $B\subset R[X]$ there exists a polynomial $F\in B$ of maximum degree $n.$ No polynomial in $R[X]$ of degree $n+1$ or more is in the submodule generated by $B.$

1.43.* If $L$ is ring-finite over $K$ (with $K,L$ fields) then $L$ is a finitely generated field extension of $K.$

Suppose $L=K[v_1,\ldots,v_n],$ with $v_1,\ldots,v_n\in L.$ Then $L=K(v_1,\ldots,v_n)$ and $L$ is a finitely generated field extension of $K.$

1.44.* Show $L=K(X)$ is a finitely generated field extension of $K,$ but is not ring-finite over $K.$

It suffices to show $L$ is not ring-finite over $K.$ Suppose $L=K[v_1,\ldots,v_n],$ with $v_1,\ldots,v_n\in L.$ Let $b\in K[X]$ be a common denominator for $v_1,\ldots,v_n.$ Let $c\in K[X]$ such that $c\not\mid b.$ Then $1/c\in L,$ so $1/c=a_1v_1+\ldots+a_nv_n$ for some $a_1,\ldots,a_n\in K.$ But $b(a_1v_1+\ldots+a_nv_n)\in K[X],$ while $b/c\not\in K[X],$ a contradiction.

1.45.* Let $R$ be a subring of $S,$ $S$ a subring of $T.$

(a) If $S=\sum Rv_i,$ $T=\sum Sw_j,$ show $T=\sum Rv_iw_j.$

Let $x\in T.$ Then $x=\sum_{j=1}^m s_jw_j,$ each $s_j\in S.$ Moreover, $s_j=\sum_{i=1}^n r_iv_i,$ each $r_i\in R.$ Thus $x=\sum Rv_iw_j.$

(b) If $S=R[v_1,\ldots,v_n]$ and $T=S[w_1,\ldots,w_m],$ then $T=R[v_1,\ldots,v_n,w_1,\ldots,w_m].$

Note that if $F\in T$ then $F=G(w_1,\ldots,w_m)$ for some $G\in S[X_1,\ldots,X_n].$ Each coefficient of $G$ is in $S=R[v_1,\ldots,v_n].$ Evaluating we have $F\in R[v_1,\ldots,v_n,w_1,\ldots,w_m].$ The converse follows by factoring.

Let $K=R(v_1,\ldots,v_n,w_1,\ldots,w_m).$ The elements $v_1,\ldots,v_n,w_,\ldots,w_m$ are contained in an extension field $L$ of $R.$ Observe that $T$ is a subfield of $L$ containing $v_1,\ldots,v_n,w_,\ldots,w_m,$ thus $K\subset T.$

Clearly $S\subset K.$ Moreover, $w_1,\ldots,w_m\in K.$ So $T\subset K.$ So $T=K.$