Chapter 2

Show that the map that associates to each \(F \in k\left[X_{1}, \ldots, X_{n}\right]\) a polynomial function in \(\mathscr{F}(V, k)\) is a ring homomorphism whose kernel is \(I(V)\).

Let \(V \subset \mathbb{A}^{n}\) be a variety. A subvariety of \(V\) is a variety \(W \subset \mathbb{A}^{n}\) that is contained in \(V\). Show that there is a natural one-to-one correspondence between algebraic subsets (resp. subvarieties, resp. points) of \(V\) and radical ideals (resp. prime ideals, resp. maximal ideals) of \(\Gamma(V)\). (See Problems 1.22, 1.38.)

Let \(W\) be a subvariety of a variety \(V\), and let \(I_{V}(W)\) be the ideal of \(\Gamma(V)\) corresponding to \(W\). (a) Show that every polynomial function on \(V\) restricts to a polynomial function on \(W\). (b) Show that the map from \(\Gamma(V)\) to \(\Gamma(W)\) defined in part (a) is a surjective homomorphism with kernel \(I_{V}(W)\), so that \(\Gamma(W)\) is isomorphic to \(\Gamma(V) / I_{V}(W)\)

Let \(V \subset \mathbb{A}^{n}\) be a nonempty variety. Show that the following are equivalent: (i) \(V\) is a point; (ii) \(\Gamma(V)=k\); (iii) \(\operatorname{dim}_{k} \Gamma(V)<\infty\).

Let \(F\) be an irreducible polynomial in \(k[X, Y]\), and suppose \(F\) is monic in \(Y\) : \(F=Y^{n}+a_{1}(X) Y^{n-1}+\cdots\), with \(n>0 .\) Let \(V=V(F) \subset \mathbb{A}^{2}\). Show that the natural homomorphism from \(k[X]\) to \(\Gamma(V)=k[X, Y] /(F)\) is one-to-one, so that \(k[X]\) may be regarded as a subring of \(\Gamma(V)\); show that the residues \(\overline{1}, \bar{Y}, \ldots, \bar{Y}^{n-1}\) generate \(\Gamma(V)\) over \(k[X]\) as a module.

If \(\varphi: V \rightarrow W\) is a polynomial map, and \(X\) is an algebraic subset of \(W\), show that \(\varphi^{-1}(X)\) is an algebraic subset of \(V\). If \(\varphi^{-1}(X)\) is irreducible, and \(X\) is contained in the image of \(\varphi\), show that \(X\) is irreducible. This gives a useful test for irreducibility.

(a) Show that \(\left\\{\left(t, t^{2}, t^{3}\right) \in \mathbb{A}^{3}(k) \mid t \in k\right\\}\) is an affine variety. (b) Show that \(V(X Z-\) \(\left.Y^{2}, Y Z-X^{3}, Z^{2}-X^{2} Y\right) \subset \mathbb{A}^{3}(\mathbb{C})\) is a variety. (Hint:: \(Y^{3}-X^{4}, Z^{3}-X^{5}, Z^{4}-Y^{5} \in I(V)\) Find a polynomial map from \(\mathbb{A}^{1}(\mathbb{C})\) onto \(V\).)

Let \(\varphi: V \rightarrow W\) be a polynomial map of affine varieties, \(V^{\prime} \subset V, W^{\prime} \subset W\) subvarieties. Suppose \(\varphi\left(V^{\prime}\right) \subset W^{\prime}\). (a) Show that \(\tilde{\varphi}\left(I_{W}\left(W^{\prime}\right)\right) \subset I_{V}\left(V^{\prime}\right)\) (see Problems 2.3). (b) Show that the restriction of \(\varphi\) gives a polynomial map from \(V^{\prime}\) to \(W^{\prime}\).

Show that the projection map pr: \(\mathbb{A}^{n} \rightarrow \mathbb{A}^{r}, n \geq r\), defined by \(\operatorname{pr}\left(a_{1}, \ldots, a_{n}\right)=\) \(\left(a_{1}, \ldots, a_{r}\right)\) is a polynomial map.

Show that the projection map pr: \(\mathbb{A}^{n} \rightarrow \mathbb{A}^{r}, n \geq r\), defined by \(\operatorname{pr}\left(a_{1}, \ldots, a_{n}\right)=\) \(\left(a_{1}, \ldots, a_{r}\right)\) is a polynomial map.

(a)] Let \(\varphi: \mathbb{A}^{1} \rightarrow V=V\left(Y^{2}-X^{3}\right) \subset \mathbb{A}^{2}\) be defined by \(\varphi(t)=\left(t^{2}, t^{3}\right)\). Show that although \(\varphi\) is a one-to-one, onto polynomial map, \(\varphi\) is not an isomorphism. (Hint:: \(\left.\tilde{\varphi}(\Gamma(V))=k\left[T^{2}, T^{3}\right] \subset k[T]=\Gamma\left(\mathbb{A}^{1}\right) .\right)\) (b) Let \(\varphi: \mathbb{A}^{1} \rightarrow V=V\left(Y^{2}-X^{2}(X+1)\right)\) be de fined by \(\varphi(t)=\left(t^{2}-1, t\left(t^{2}-1\right)\right)\). Show that \(\varphi\) is one-to-one and onto, except that \(\varphi(\pm 1)=(0,0)\)

\(\mathrm{A}\) set \(V \subset \mathbb{A}^{n}(k)\) is called a linear subvariety of \(\mathbb{A}^{n}(k)\) if \(V=V\left(F_{1}, \ldots, F_{r}\right)\) for some polynomials \(F_{i}\) of degree 1. (a) Show that if \(T\) is an affine change of coordinates on \(\mathbb{A}^{n}\), then \(V^{T}\) is also a linear subvariety of \(A^{n}(k)\). (b) If \(V \neq \varnothing\), show that there is an affine change of coordinates \(T\) of \(\mathbb{A}^{n}\) such that \(V^{T}=V\left(X_{m+1}, \ldots, X_{n}\right)\).

Let \(P=\left(a_{1}, \ldots, a_{n}\right), Q=\left(b_{1}, \ldots, b_{n}\right)\) be distinct points of \(\mathbb{A}^{n}\). The line through \(P\) and \(Q\) is defined to be \(\left.\left\\{a_{1}+t\left(b_{1}-a_{1}\right), \ldots, a_{n}+t\left(b_{n}-a_{n}\right)\right) \mid t \in k\right\\}\). (a) Show that if \(L\) is the line through \(P\) and \(Q\), and \(T\) is an affine change of coordinates, then \(T(L)\) is the line through \(T(P)\) and \(T(Q)\). (b) Show that a line is a linear subvariety of dimension 1, and that a linear subvariety of dimension 1 is the line through any two of its points. (c) Show that, in \(\mathbb{A}^{2}\), a line is the same thing as a hyperplane. (d) Let \(P\), \(P^{\prime} \in \mathbb{A}^{2}, L_{1}, L_{2}\) two distinct lines through \(P, L_{1}^{\prime}, L_{2}^{\prime}\) distinct lines through \(P^{\prime}\). Show that there is an affine change of coordinates \(T\) of \(\mathbb{A}^{2}\) such that \(T(P)=P^{\prime}\) and \(T\left(L_{i}\right)=L_{i}^{\prime}, i=1,2\).

Let \(k=\mathbb{C}\). Give \(\mathbb{A}^{n}(\mathbb{C})=\mathbb{C}^{n}\) the usual topology (obtained by identifying \(\mathbb{C}\) with \(\mathbb{R}^{2}\), and hence \(\mathbb{C}^{n}\) with \(\mathbb{R}^{2 n}\) ). Recall that a topological space \(X\) is path-connected if for any \(P, Q \in X\), there is a continuous mapping \(\gamma:[0,1] \rightarrow X\) such that \(\gamma(0)=P, \gamma(1)=Q\). (a) Show that \(\mathbb{C} \backslash S\) is path-connected for any finite set \(S\). (b) Let \(V\) be an algebraic set in \(\mathbb{A}^{n}(\mathbb{C})\). Show that \(\mathbb{A}^{n}(\mathbb{C}) \backslash V\) is path-connected. (Hint:: If \(P, Q \in \mathbb{A}^{n}(\mathbb{C}) \backslash V\), let \(L\) be the line through \(P\) and \(Q\). Then \(L \cap V\) is finite, and \(L\) is isomorphic to \(A^{1}\) (C).)

Let \(V=V\left(Y^{2}-X^{2}(X+1)\right) \subset \mathbb{A}^{2}\), and \(\bar{X}, \bar{Y}\) the residues of \(X, Y\) in \(\Gamma(V)\); let \(z=\bar{Y} / \bar{X} \in k(V)\). Find the pole sets of \(z\) and of \(z^{2}\)

Let \(\mathscr{O}_{P}(V)\) be the local ring of a variety \(V\) at a point \(P\). Show that there is a natural one-to-one correspondence between the prime ideals in \(\mathscr{O}_{P}(V)\) and the subvarieties of \(V\) that pass through \(P\). (Hint:: If \(I\) is prime in \(\mathscr{O}_{P}(V), I \cap \Gamma(V)\) is prime in \(\Gamma(V)\), and \(I\) is generated by \(I \cap \Gamma(V)\); use Problem 2.2.)

Let \(f\) be a rational function on a variety \(V\). Let \(U=\\{P \in V \mid f\) is defined at P\\}. Then \(f\) defines a function from \(U\) to \(k\). Show that this function determines \(f\) uniquely. So a rational function may be considered as a type of function, but only on the complement of an algebraic subset of \(V\), not on \(V\) itself.

Let \(\varphi: V \rightarrow W\) be a polynomial map of affine varieties, \(\tilde{\varphi}: \Gamma(W) \rightarrow \Gamma(V)\) the induced map on coordinate rings. Suppose \(P \in V, \varphi(P)=Q\). Show that \(\tilde{\varphi}\) extends uniquely to a ring homomorphism (also written \(\tilde{\varphi}\) ) from \(\mathscr{O}_{Q}(W)\) to \(\mathscr{O}_{P}(V)\). (Note that \(\tilde{\varphi}\) may not extend to all of \(k(W)\).) Show that \(\tilde{\varphi}\left(\mathrm{m}_{Q}(W)\right) \subset \mathfrak{m}_{P}(V)\).

Let \(T: \mathbb{A}^{n} \rightarrow \mathbb{A}^{n}\) be an affine change of coordinates, \(T(P)=Q\). Show that \(\tilde{T}: \mathscr{O}_{Q}\left(\mathbb{A}^{n}\right) \rightarrow \mathscr{O}_{P}\left(\mathbb{A}^{n}\right)\) is an isomorphism. Show that \(\tilde{T}\) induces an isomorphism from \(\mathscr{O}_{Q}(V)\) to \(\mathscr{O}_{P}\left(V^{T}\right)\) if \(P \in V^{T}\), for \(V\) a subvariety of \(\mathbb{A}^{n}\).

Show that the order function on \(K\) is independent of the choice of uniformizing parameter.

Let \(V=\mathbb{A}^{1}, \Gamma(V)=k[X], K=k(V)=k(X)\). (a) For each \(a \in k=V\), show that \(\mathscr{O}_{a}(V)\) is a DVR with uniformizing parameter \(t=X-a\). (b) Show that \(\mathscr{O}_{\infty}=\\{F / G \in\) \(k(X) \mid \operatorname{deg}(G) \geq \operatorname{deg}(F)\\}\) is also a DVR, with uniformizing parameter \(t=1 / X\).

Let \(p \in \mathbb{Z}\) be a prime number. Show that \(\\{r \in Q \mid r=a / b, a, b \in \mathbb{Z}, p\) doesn't divide \(b\\}\) is a DVR with quotient field \(Q\).

Let \(R\) be a DVR with quotient field \(K\); let \(\mathfrak{m}\) be the maximal ideal of \(R\). (a) Show that if \(z \in K, z \notin R\), then \(z^{-1} \in m\). (b) Suppose \(R \subset S \subset K\), and \(S\) is also a DVR. Suppose the maximal ideal of \(S\) contains \(\mathrm{m}\). Show that \(S=R\).

An order function on a field \(K\) is a function \(\varphi\) from \(K\) onto \(\mathbb{Z} \cup\\{\infty\\}\), satisfying: (i) \(\varphi(a)=\infty\) if and only if \(a=0\). (ii) \(\varphi(a b)=\varphi(a)+\varphi(b)\). (iii) \(\varphi(a+b) \geq \min (\varphi(a), \varphi(b))\). Show that \(R=\\{z \in K \mid \varphi(z) \geq 0\\}\) is a DVR with maximal ideal \(\mathfrak{m}=\\{z \mid \varphi(z)>0\\}\), and quotient field \(K\). Conversely, show that if \(R\) is a DVR with quotient field \(K\), then the function ord: \(K \rightarrow \mathbb{Z} \cup\\{\infty\\}\) is an order function on \(K\). Giving a DVR with quotient field \(K\) is equivalent to defining an order function on \(K\).

Let \(R\) be a DVR with quotient field \(K\), ord the order function on \(K\). (a) If \(\operatorname{ord}(a)<\operatorname{ord}(b)\), show that ord \((a+b)=\operatorname{ord}(a)\). (b) If \(a_{1}, \ldots, a_{n} \in K\), and for some \(i, \operatorname{ord}\left(a_{i}\right)<\operatorname{ord}\left(a_{j}\right)\) (all \(j \neq i\) ), then \(a_{1}+\cdots+a_{n} \neq 0\).

Let \(R\) be a DVR with maximal ideal \(\mathfrak{m}\), and quotient field \(K\), and suppose a field \(k\) is a subring of \(R\), and that the composition \(k \rightarrow R \rightarrow R / \mathrm{m}\) is an isomorphism of \(k\) with \(R / m\) (as for example in Problem 2.24). Verify the following assertions: (a) For any \(z \in R\), there is a unique \(\lambda \in k\) such that \(z-\lambda \in \mathrm{m}\). (b) Let \(t\) be a uniformizing parameter for \(R, z \in R\). Then for any \(n \geq 0\) there are unique \(\lambda_{0}, \lambda_{1}, \ldots, \lambda_{n} \in k\) and \(z_{n} \in R\) such that \(z=\lambda_{0}+\lambda_{1} t+\lambda_{2} t^{2}+\cdots+\lambda_{n} t^{n}+z_{n} t^{n+1}\).

Let \(k\) be a field. The ring of formal power series over \(k\), written \(k[[X]]\), is defined to be \(\left\\{\sum_{i=0}^{\infty} a_{i} X^{i} \mid a_{i} \in k\right\\}\). (As with polynomials, a rigorous definition is best given in terms of sequences \(\left(a_{0}, a_{1}, \ldots\right)\) of elements in \(k\); here we allow an infinite number of nonzero terms.) Define the sum by \(\sum a_{i} X^{i}+\sum b_{i} X^{i}=\Sigma\left(a_{i}+b_{i}\right) X^{i}\), and the product by \(\left(\sum a_{i} X^{i}\right)\left(\sum b_{i} X^{i}\right)=\sum c_{i} X^{i}\), where \(c_{i}=\sum_{j+k=i} a_{j} b_{k}\). Show that \(k[[X]]\) is a ring containing \(k[X]\) as a subring. Show that \(k[[X]]\) is a DVR with uniformizing parameter \(X\). Its quotient field is denoted \(k((X))\).

Factor \(Y^{3}-2 X Y^{2}+2 X^{2} Y+X^{3}\) into linear factors in \(\mathbb{C}[X, Y]\)

Suppose \(F, G \in k\left[X_{1}, \ldots, X_{n}\right]\) are forms of degree \(r, r+1\) respectively, with no common factors ( \(k\) a field). Show that \(F+G\) is irreducible.

(a) Show that there are \(d+1\) monomials of degree \(d\) in \(R[X, Y]\), and \(1+2+\) \(\cdots+(d+1)=(d+1)(d+2) / 2\) monomials of degree \(d\) in \(R[X, Y, Z]\). (b) Let \(V(d, n)=\) \(\left\\{\right.\) forms of degree \(d\) in \(\left.k\left[X_{1}, \ldots, X_{n}\right]\right\\}, k\) a field. Show that \(V(d, n)\) is a vector space over \(k\), and that the monomials of degree \(d\) form a basis. So \(\operatorname{dim} V(d, 1)=1 ; \operatorname{dim} V(d, 2)=\) \(d+1 ; \operatorname{dim} V(d, 3)=(d+1)(d+2) / 2 .\) (c) Let \(L_{1}, L_{2}, \ldots\) and \(M_{1}, M_{2}, \ldots\) be sequences of nonzero linear forms in \(k[X, Y]\), and assume no \(L_{i}=\lambda M_{j}, \lambda \in k .\) Let \(A_{i j}=\) \(L_{1} L_{2} \ldots L_{i} M_{1} M_{2} \ldots M_{j}, i, j \geq 0\left(A_{00}=1\right)\). Show that \(\left\\{A_{i j} \mid i+j=d\right\\}\) forms a basis for \(V(d, 2)\).

With the above notation, show that \(\operatorname{dim} V(d, n)=\left(\begin{array}{c}d+n-1 \\ n-1\end{array}\right)\), the binomial coefficient.

What are the additive and multiplicative identities in \(\prod R_{i}\) ? Is the map from \(R_{i}\) to \(\Pi R_{j}\) taking \(a_{i}\) to \(\left(0, \ldots, a_{i}, \ldots, 0\right.\) ) a ring homomorphism?

Prove the following relations among ideals \(I_{i}, J\), in a ring \(R\) : (a) \(\left(I_{1}+I_{2}\right) J=I_{1} J+I_{2} J\). (b) \(\left(I_{1} \cdots I_{N}\right)^{n}=I_{1}^{n} \cdots I_{N}^{n}\)

(a) Suppose \(I, J\) are comaximal ideals in \(R\). Show that \(I+J^{2}=R\). Show that \(I^{m}\) and \(J^{n}\) are comaximal for all \(m, n .\) (b) Suppose \(I_{1}, \ldots, I_{N}\) are ideals in \(R\), and \(I_{i}\) and \(J_{i}=\bigcap_{j \neq i} I_{j}\) are comaximal for all \(i\). Show that \(I_{1}^{n} \cap \cdots \cap I_{N}^{n}=\left(I_{1} \cdots I_{N}\right)^{n}=\) \(\left(I_{1} \cap \cdots \cap I_{N}\right)^{n}\) for all \(n\)

Let \(I, J\) be ideals in a ring \(R\). Suppose \(I\) is finitely generated and \(I \subset \operatorname{Rad}(J)\). Show that \(I^{n} \subset J\) for some \(n .\)

(a) Let \(I \subset J\) be ideals in a ring \(R\). Show that there is a natural ring homomorphism from \(R / I\) onto \(R / J\). (b) Let \(I\) be an ideal in a ring \(R, R\) a subring of a ring \(S\). Show that there is a natural ring homomorphism from \(R / I\) to \(S / I S\).

Let \(P=(0, \ldots, 0) \in \mathbb{A}^{n}, \mathscr{O}=\mathscr{O}_{P}\left(\mathbb{A}^{n}\right), \mathfrak{m}=\mathfrak{m}_{P}\left(\mathbb{A}^{n}\right)\). Let \(I \subset k\left[X_{1}, \ldots, X_{n}\right]\) be the ideal generated by \(X_{1}, \ldots, X_{n}\). Show that \(I \mathscr{O}=m\), so \(I^{r} \mathscr{O}=m^{r}\) for all \(r\).

Let \(V\) be a variety in \(\mathbb{A}^{n}, I=I(V) \subset k\left[X_{1}, \ldots, X_{n}\right], P \in V\), and let \(J\) be an ideal of \(k\left[X_{1}, \ldots, X_{n}\right]\) that contains \(I .\) Let \(J^{\prime}\) be the image of \(J\) in \(\Gamma(V)\). Show that there is a natural homomorphism \(\varphi\) from \(\mathscr{O}_{P}\left(\mathbb{A}^{n}\right) / J \mathscr{O}_{P}\left(\mathbb{A}^{n}\right)\) to \(\mathscr{O}_{P}(V) / J^{\prime} \mathscr{O}_{P}(V)\), and that \(\varphi\) is an isomorphism. In particular, \(\mathscr{O}_{P}\left(\mathbb{A}^{n}\right) / I \mathscr{O}_{P}\left(\mathbb{A}^{n}\right)\) is isomorphic to \(\mathscr{O}_{P}(V)\).

Show that ideals \(I, J \subset k\left[X_{1}, \ldots, X_{n}\right]\) ( \(k\) algebraically closed) are comaximal if and only if \(V(I) \cap V(J)=\phi\).

Let \(I=(X, Y) \subset k[X, Y]\). Show that \(\operatorname{dim}_{k}\left(k[X, Y] / I^{n}\right)=1+2+\cdots+n=\frac{n(n+1)}{2}\).

Suppose \(R\) is a ring containing \(k\), and \(R\) is finite dimensional over \(k\). Show that \(R\) is isomorphic to a direct product of local rings.

Verify that for any \(R\)-module homomorphism \(\varphi: M \rightarrow M^{\prime}, \operatorname{Ker}(\varphi)\) and \(\operatorname{Im}(\varphi)\) are submodules of \(M\) and \(M^{\prime}\) respectively. Show that $$ 0 \longrightarrow \operatorname{Ker}(\varphi) \longrightarrow M \stackrel{\varphi}{\longrightarrow} \operatorname{Im}(\varphi) \longrightarrow 0 $$ is exact.

(a) Let \(N\) be a submodule of \(M, \pi: M \rightarrow M / N\) the natural homomorphism. Suppose \(\varphi: M \rightarrow M^{\prime}\) is a homomorphism of \(R\)-modules, and \(\varphi(N)=0\). Show that there is a unique homomorphism \(\bar{\varphi}: M / N \rightarrow M^{\prime}\) such that \(\bar{\varphi} \circ \pi=\varphi .\) (b) If \(N\) and \(P\) are submodules of a module \(M\), with \(P \subset N\), then there are natural homomorphisms from \(M / P\) onto \(M / N\) and from \(N / P\) into \(M / P .\) Show that the resulting sequence $$ 0 \longrightarrow N / P \longrightarrow M / P \longrightarrow M / N \longrightarrow 0 $$ is exact ("Second Noether Isomorphism Theorem"). (c) Let \(U \subset W \subset V\) be vector spaces, with \(V / U\) finite-dimensional. Then \(\operatorname{dim} V / U=\operatorname{dim} V / W+\operatorname{dim} W / U\). (d) If \(J \subset I\) are ideals in a ring \(R\), there is a natural exact sequence of \(R\)-modules: $$ 0 \longrightarrow I / J \longrightarrow R / J \longrightarrow R / I \longrightarrow 0 $$ (e) If \(\mathscr{O}\) is a local ring with maximal ideal \(\mathrm{m}\), there is a natural exact sequence of \(\mathscr{O}\)-modules $$ 0 \longrightarrow \mathrm{m}^{n} / \mathrm{m}^{n+1} \longrightarrow \mathscr{O} / \mathrm{m}^{n+1} \longrightarrow \mathscr{O} / \mathrm{m}^{n} \longrightarrow 0 $$

Let \(R\) be a DVR satisfying the conditions of Problem \(2.30\). Then \(\mathfrak{m}^{n} / \mathrm{m}^{n+1}\) is an \(R\)-module, and so also a \(k\)-module, since \(k \subset R\). (a) Show that \(\operatorname{dim}_{k}\left(\mathfrak{m}^{n} / \mathrm{m}^{n+1}\right)=1\) for all \(n \geq 0\). (b) Show that \(\operatorname{dim}_{k}\left(R / \mathfrak{m}^{n}\right)=n\) for all \(n>0\). (c) Let \(z \in R\). Show that \(\operatorname{ord}(z)=n\) if \((z)=\mathfrak{m}^{n}\), and hence that \(\operatorname{ord}(z)=\operatorname{dim}_{k}(R /(z))\).

Let \(0 \longrightarrow V_{1} \longrightarrow \cdots \longrightarrow V_{n} \longrightarrow 0\) be an exact sequence of finite-dimensional vector spaces. Show that \(\sum(-1)^{i} \operatorname{dim}\left(V_{i}\right)=0\).

Let \(N, P\) be submodules of a module \(M\). Show that the subgroup \(N+P=\) \(\\{n+p \mid n \in N, p \in P\\}\) is a submodule of \(M\). Show that there is a natural \(R\)-module isomorphism of \(N / N \cap P\) onto \(N+P / P\) (“First Noether Isomorphism Theorem").

Let \(N, P\) be submodules of a module \(M\). Show that the subgroup \(N+P=\) \(\\{n+p \mid n \in N, p \in P\\}\) is a submodule of \(M\). Show that there is a natural \(R\)-module isomorphism of \(N / N \cap P\) onto \(N+P / P\) (“First Noether Isomorphism Theorem").

What does \(M\) being free on \(m_{1}, \ldots, m_{n}\) say in terms of the elements of \(M\) ?

Let \(F=X^{n}+a_{1} X^{n-1}+\cdots+a_{n}\) be a monic polynomial in \(R[X]\). Show that \(R[X] /(F)\) is a free \(R\)-module with basis \(\overline{1}, \bar{X}, \ldots, \bar{X}^{n-1}\), where \(\bar{X}\) is the residue of \(X\).

Show that a subset \(X\) of a module \(M\) generates \(M\) if and only if the homomorphism \(M_{X} \rightarrow M\) is onto. Every module is isomorphic to a quotient of a free module.