Chapter 7

Let \(R\) be a ring. For additive subgroups \(A\) and \(B\) of \(R\), we define their ring-theoretic product \(A B\) as the set of all elements of \(R\) that can be expressed as $$ a_{1} b_{1}+\cdots+a_{k} b_{k} $$ for some \(a_{1}, \ldots, a_{k} \in A\) and \(b_{1}, \ldots, b_{k} \in B ;\) by definition, this set includes the "empty sum" \(0_{R} .\) Show that for all additive subgroups \(A, B,\) and \(C\) of \(R\) : (a) \(A B\) is also an additive subgroup of \(R\); (b) \(A B=B A\) (c) \(A(B C)=(A B) C\) (d) \(A(B+C)=A B+A C\)

Let \(R\) be a ring, and let \(a, b \in R\) such that \(a b \neq 0 .\) Show that \(a b\) is a zero divisor if and only if \(a\) is a zero divisor or \(b\) is a zero divisor.

Suppose that \(R\) is a non-trivial ring in which the cancellation law holds in general: for all \(a, b, c \in R,\) if \(a \neq 0\) and \(a b=a c,\) then \(b=c .\) Show that \(R\) is an integral domain.

Let \(R\) be a ring of characteristic \(m>0,\) and let \(n\) be an integer. Show that: (a) if \(\operatorname{gcd}(n, m)=1,\) then \(n \cdot 1_{R}\) is a unit; (b) if \(1<\operatorname{gcd}(n, m)

Let \(D\) be an integral domain, \(m \in \mathbb{Z},\) and \(a \in D .\) Show that \(m a=0\) if and only if \(m\) is a multiple of the characteristic of \(D\) or \(a=0\)

Show that for all \(n \geq 1,\) and for all \(a, b \in \mathbb{Z}_{n},\) if \(a \mid b\) and \(b \mid a\), then \(a r=b\) for some \(r \in \mathbb{Z}_{n}^{*}\). Hint: this result does not follow from part (i) of Theorem \(7.4,\) as we allow \(a\) and \(b\) to be zero divisors here; first consider the case where \(n\) is a prime power.

Show that the ring \(\mathcal{F}\) of arithmetic functions defined in Example 7.6 is an integral domain.

This exercise depends on results in \(\$ 6.6 .\) Using the fundamental theorem of finite abelian groups, show that the additive group of a finite field of characteristic \(p\) and cardinality \(p^{w}\) is isomorphic to \(\mathbb{Z}_{p}^{\times w}\)

Show that if \(S\) is a subring of a ring \(R,\) then a set \(T \subseteq S\) is a subring of \(R\) if and only if \(T\) is a subring of \(S\).

Show that if \(S\) and \(T\) are subrings of \(R\), then so is \(S \cap T\).

Let \(S_{1}\) be a subring of \(R_{1},\) and \(S_{2}\) a subring of \(R_{2}\). Show that \(S_{1} \times S_{2}\) is a subring of \(R_{1} \times R_{2}\)

Show that the set \(\mathbb{Q}[i]\) of complex numbers of the form \(a+b i\), with \(a, b \in \mathbb{Q},\) is a subfield of \(\mathbb{C}\).

Consider the ring \(\operatorname{Map}(\mathbb{R}, \mathbb{R})\) of functions \(f: \mathbb{R} \rightarrow \mathbb{R},\) with addition and multiplication defined point-wise. (a) Show that \(\operatorname{Map}(\mathbb{R}, \mathbb{R})\) is not an integral domain, and that \(\operatorname{Map}(\mathbb{R}, \mathbb{R})^{*}\) consists of those functions that never vanish. (b) Let \(a, b \in \operatorname{Map}(\mathbb{R}, \mathbb{R}) .\) Show that if \(a \mid b\) and \(b \mid a,\) then \(a r=b\) for some \(r \in \operatorname{Map}(\mathbb{R}, \mathbb{R})^{*}\) (c) Let \(\mathcal{C}\) be the subset of \(\operatorname{Map}(\mathbb{R}, \mathbb{R})\) of continuous functions. Show that \(\mathcal{C}\) is a subring of \(\operatorname{Map}(\mathbb{R}, \mathbb{R}),\) and that all functions in \(\mathcal{C}^{*}\) are either everywhere positive or everywhere negative. (d) Find elements \(a, b \in \mathcal{C},\) such that in the ring \(\mathcal{C},\) we have \(a \mid b\) and \(b \mid a,\) yet there is no \(r \in \mathcal{C}^{*}\) such that \(a r=b\).

Let \(D\) be an infinite integral domain, and let \(g, h \in D[X] .\) Show that if \(g(x)=h(x)\) for all \(x \in D,\) then \(g=h .\) Thus, for an infinite integral domain \(D,\) there is a one-to-one correspondence between polynomials over \(D\) and polynomial functions on \(D\).

Let \(F\) be a field. (a) Show that for all \(b \in F,\) we have \(b^{2}=1\) if and only if \(b=\pm 1\). (b) Show that for all \(a, b \in F,\) we have \(a^{2}=b^{2}\) if and only if \(a=\pm b\). (c) Show that the familiar quadratic formula holds for \(F\), assuming \(F\) has characteristic other than 2 , so that \(2_{F} \neq 0_{F}\). That is, for all \(a, b, c \in F\) with \(a \neq 0,\) the polynomial \(g:=a X^{2}+b X+c \in F[X]\) has a root in \(F\) if and only if there exists \(e \in F\) such that \(e^{2}=d,\) where \(d\) is the discriminant of \(g,\) defined as \(d:=b^{2}-4 a c,\) and in this case the roots of \(g\) are \((-b \pm e) / 2 a\)

Let \(R\) be a ring, let \(g \in R[X],\) with \(\operatorname{deg}(g)=k \geq 0,\) and let \(x\) be an element of \(R\). Show that: (a) there exist an integer \(m,\) with \(0 \leq m \leq k,\) and a polynomial \(q \in R[X],\) such that $$ g=(X-x)^{m} q \text { and } q(x) \neq 0 $$ and moreover, the values of \(m\) and \(q\) are uniquely determined; (b) if we evaluate \(g\) at \(X+x,\) we have $$ g(X+x)=\sum_{i=0}^{k} b_{i} X^{i} $$ where \(b_{0}=\cdots=b_{m-1}=0\) and \(b_{m}=q(x) \neq 0\)

Let \(D\) be an integral domain, and suppose that \(g \in D[X]\), with \(\operatorname{deg}(g)=k \geq 0 .\) Show that $$ \sum_{x \in D} m_{x}(g) \leq k $$

Let \(D\) be an integral domain, let \(g, h \in D[X],\) and let \(x \in D\). Show that \(m_{x}(g h)=m_{x}(g)+m_{x}(h)\).

Let \(R\) be a ring, and consider the ring of multi-variate polynomials \(R\left[X_{1}, \ldots, X_{n}\right]\). For \(m \geq 0,\) define \(H_{m}\) to be the subset of polynomials that can be expressed as \(a_{1} \mu_{1}+\cdots+a_{k} \mu_{k},\) where each \(a_{i}\) belongs to \(R\) and each \(\mu_{i}\) is a monomial of total degree \(m\) (by definition, \(H_{m}\) includes the zero polynomial, and \(\left.H_{0}=R\right) .\) Polynomials that belong to \(H_{m}\) for some \(m\) are called homogeneous polynomials. Show that: (a) if \(g, h \in H_{m},\) then \(g+h \in H_{m}\) (b) if \(g \in H_{\ell}\) and \(h \in H_{m},\) then \(g h \in H_{\ell+m}\); (c) every non-zero polynomial \(g\) can be expressed uniquely as \(g_{0}+\cdots+g_{d}\), where \(g_{i} \in H_{i}\) for \(i=0, \ldots, d, g_{d} \neq 0,\) and \(d=\operatorname{Deg}(g)\) (d) for all polynomials \(g, h,\) we have \(\operatorname{Deg}(g h) \leq \operatorname{Deg}(g)+\operatorname{Deg}(h),\) and if \(R\) is an integral domain, then \(\operatorname{Deg}(g h)=\operatorname{Deg}(g)+\operatorname{Deg}(h)\).

Suppose that \(D\) is an integral domain, and \(g, h\) are non-zero, multi-variate polynomials over \(D\) such that \(g h\) is homogeneous. Show that \(g\) and \(h\) are also homogeneous.

Show that if \(F\) is a field, then the only ideals of \(F\) are \(\left\\{0_{F}\right\\}\) and \(F\)

Let \(a, b\) be elements of a ring \(R\). Show that $$ a \mid b \Longleftrightarrow b \in a R \Leftrightarrow b R \subseteq a R $$

Let \(R\) be a ring. Show that if \(I\) is a non-empty subset of \(R[X]\) that is closed under addition, multiplication by elements of \(R,\) and multiplication by \(X,\) then \(I\) is an ideal of \(R[X]\).

Let \(I\) be an ideal of \(R\), and \(S\) a subring of \(R\). Show that \(I \cap S\) is an ideal of \(S\)

Let \(I\) be an ideal of \(R,\) and \(S\) a subring of \(R .\) Show that \(I+S\) is a subring of \(R,\) and that \(I\) is an ideal of \(I+S\).

Let \(I_{1}\) be an ideal of \(R_{1},\) and \(I_{2}\) an ideal of \(R_{2} .\) Show that \(I_{1} \times I_{2}\) is an ideal of \(R_{1} \times R_{2}\)

Write down the multiplication table for \(\mathbb{Z}_{2}[X] /\left(X^{2}+X\right)\). Is this a field?

Let \(I\) be an ideal of a ring \(R,\) and let \(x\) and \(y\) be elements of \(R\) with \(x \equiv y(\bmod I)\). Let \(g \in R[X]\). Show that \(g(x) \equiv g(y)(\bmod I)\).

Let \(R\) be a ring, and fix \(x_{1}, \ldots, x_{n} \in R\). Let $$ I:=\left\\{g \in R\left[X_{1}, \ldots, X_{n}\right]: g\left(x_{1}, \ldots, x_{n}\right)=0\right\\} $$ Show that \(I\) is an ideal of \(R\left[X_{1}, \ldots, X_{n}\right],\) and that \(I=\left(X_{1}-x_{1}, \ldots, X_{n}-x_{n}\right)\)

Let \(p\) be a prime, and consider the ring \(\mathbb{Q}^{(p)}\) (see Example 7.26). Show that every non-zero ideal of \(\mathbb{Q}^{(p)}\) is of the form \(\left(p^{i}\right),\) for some uniquely determined integer \(i \geq 0\)

Let \(p\) be a prime. Show that in the ring \(\mathbb{Z}[X],\) the ideal \((X, p)\) is not a principal ideal.

Let \(F\) be a field. Show that in the ring \(F[X, Y],\) the ideal \((X, Y)\) is not a principal ideal.

Let \(R\) be a ring, and let \(\left\\{I_{i}\right\\}_{i=0}^{\infty}\) be a sequence of ideals of \(R\) such that \(I_{i} \subseteq I_{i+1}\) for all \(i=0,1,2, \ldots .\) Show that the union \(\bigcup_{i=0}^{\infty} I_{i}\) is also an ideal of \(R .\)

Let \(R\) be a ring. An ideal \(I\) of \(R\) is called prime if \(I \subsetneq R\) and if for all \(a, b \in R, a b \in I\) implies \(a \in I\) or \(b \in I .\) An ideal \(I\) of \(R\) is called maximal if \(I \subsetneq R\) and there are no ideals \(J\) of \(R\) such that \(I \subsetneq J \subsetneq R\). Show that: (a) an ideal \(I\) of \(R\) is prime if and only if \(R / I\) is an integral domain; (b) an ideal \(I\) of \(R\) is maximal if and only if \(R / I\) is a field; (c) all maximal ideals of \(R\) are also prime ideals.

This exercise explores some examples of prime and maximal ideals. Show that: (a) in the ring \(\mathbb{Z},\) the ideal \\{0\\} is prime but not maximal, and that the maximal ideals are precisely those of the form \(p \mathbb{Z},\) where \(p\) is prime; (b) in an integral domain \(D,\) the ideal \\{0\\} is prime, and this ideal is maximal if and only if \(D\) is a field; (c) if \(p\) is a prime, then in the ring \(\mathbb{Z}[X],\) the ideal \((X, p)\) is maximal, while the ideals \((X)\) and \((p)\) are prime, but not maximal; (d) if \(F\) is a field, then in the ring \(F[X, Y],\) the ideal \((X, Y)\) is maximal, while the ideals \((X)\) and \((Y)\) are prime, but not maximal.

It is a fact that every non-trivial ring \(R\) contain at least one maximal ideal. Showing this in general requires some fancy set-theoretic notions. This exercise develops a simple proof in the case where \(R\) is countable (see \(\S\) A3). (a) Show that if \(R\) is non-trivial but finite, then it contains a maximal ideal. (b) Assume that \(R\) is countably infinite, and let \(a_{1}, a_{2}, a_{3}, \ldots\) be an enumeration of the elements of \(R\). Define a sequence of ideals \(I_{0}, I_{1}, I_{2}, \ldots,\) as follows. Set \(I_{0}:=\left\\{0_{R}\right\\},\) and for each \(i \geq 0,\) define $$ I_{i+1}:=\left\\{\begin{array}{ll} I_{i}+a_{i} R & \text { if } I_{i}+a_{i} R \subsetneq R \\ I_{i} & \text { otherwise } \end{array}\right. $$ Finally, set \(I:=\bigcup_{i=0}^{\infty} I_{i},\) which by Exercise 7.37 is an ideal of \(R\). Show that \(I\) is a maximal ideal of \(R\). Hint: first, show that \(I \subsetneq R\) by assuming that \(1_{R} \in I\) and deriving a contradiction; then, show that \(I\) is maximal by assuming that for some \(i=1,2, \ldots,\) we have \(I \subsetneq I+a_{i} R \subsetneq R,\) and deriving a contradiction.

Let \(R\) be a ring, and let \(I\) and \(J\) be ideals of \(R\). With the ringtheoretic product as defined in Exercise \(7.2,\) show that: (a) \(I J\) is an ideal; (b) if \(I\) and \(J\) are principal ideals, with \(I=a R\) and \(J=b R,\) then \(I J=a b R\), and so is also a principal ideal; (c) \(I J \subseteq I \cap J\) (d) if \(I+J=R,\) then \(I J=I \cap J\).

Let \(M\) be a maximal ideal of a ring \(R,\) and let \(a, b \in R\). Show that if \(a b \in M^{2}\) and \(b \notin M,\) then \(a \in M^{2} .\) Here, \(M^{2}:=M M,\) the ring-theoretic product.

Let \(F\) be a field, let \(f \in F[X, Y],\) and let \(E:=F[X, Y] /(f)\) Define \(V(f):=\\{(x, y) \in F \times F: f(x, y)=0\\}\) (a) Every element \(\alpha\) of \(E\) naturally defines a function from \(V(f)\) to \(F,\) as follows: if \(\alpha=[g]_{f},\) with \(g \in F[X, Y],\) then for \(P=(x, y) \in V(f),\) we define \(\alpha(P):=g(x, y) .\) Show that this definition is unambiguous, that is, \(g \equiv h(\bmod f)\) implies \(g(x, y)=h(x, y)\) (b) For \(P=(x, y) \in V(f),\) define \(M_{P}:=\\{\alpha \in E: \alpha(P)=0\\} .\) Show that \(M_{P}\) is a maximal ideal of \(E,\) and that \(M_{P}=\mu E+v E,\) where \(\mu:=[X-x]_{f}\) and \(v:=[Y-y]_{f}\)

Continuing with the previous exercise, now assume that the characteristic of \(F\) is \(\operatorname{not} 2,\) and that \(f=Y^{2}-\phi,\) where \(\phi \in F[X]\) is a non-zero polynomial with no multiple roots in \(F\) (see definitions after Exercise 7.18 ). (a) Show that if \(P=(x, y) \in V(f),\) then so is \(\bar{P}:=(x,-y),\) and that \(P=\bar{P} \Longleftrightarrow y=0 \Longleftrightarrow \phi(x)=0\) (b) Let \(P=(x, y) \in V(f)\) and \(\mu:=[X-x]_{f} \in E .\) Show that \(\mu E=M_{P} M_{\bar{P}}\) (the ring-theoretic product). Hint: use Exercise \(7.43,\) and treat the cases \(P=\bar{P}\) and \(P \neq \bar{P}\) separately.

Let \(R\) be a ring, and \(I\) an ideal of \(R .\) Define \(\operatorname{Rad}(I)\) to be the set of all \(a \in R\) such that \(a^{n} \in I\) for some positive integer \(n\). (a) Show that \(\operatorname{Rad}(I)\) is an ideal of \(R\) containing \(I\). Hint: show that if \(a^{n} \in I\) and \(b^{m} \in I,\) then \((a+b)^{n+m} \in I .\) (b) Show that if \(R=\mathbb{Z}\) and \(I=(d),\) where \(d=p_{1}^{e_{1}} \cdots p_{r}^{e_{r}}\) is the prime factorization of \(d,\) then \(\operatorname{Rad}(I)=\left(p_{1} \cdots p_{r}\right)\)

Show that if \(\rho: F \rightarrow R\) is a ring homomorphism from a field \(F\) into a ring \(R,\) then either \(R\) is trivial or \(\rho\) is injective. Hint: use Exercise \(7.25 .\)

Verify that the "is isomorphic to" relation on rings is an equivalence relation; that is, for all rings \(R_{1}, R_{2}, R_{3},\) we have: (a) \(R_{1} \cong R_{1}\) (b) \(R_{1} \cong R_{2}\) implies \(R_{2} \cong R_{1}\) (c) \(R_{1} \cong R_{2}\) and \(R_{2} \cong R_{3}\) implies \(R_{1} \cong R_{3}\).

Let \(\rho_{i}: R_{i} \rightarrow R_{i}^{\prime},\) for \(i=1, \ldots, k,\) be ring homomorphisms. Show that the map $$ \begin{aligned} \rho: \quad R_{1} \times \cdots \times R_{k} & \rightarrow R_{1}^{\prime} \times \cdots \times R_{k}^{\prime} \\ \left(a_{1}, \ldots, a_{k}\right) & \mapsto\left(\rho_{1}\left(a_{1}\right), \ldots, \rho_{k}\left(a_{k}\right)\right) \end{aligned} $$ is a ring homomorphism.

Let \(\rho: R \rightarrow R^{\prime}\) be a ring homomorphism, and let \(a \in R\). Show that \(\rho(a \boldsymbol{R})=\rho(a) \rho(\boldsymbol{R})\)

Let \(\rho: R \rightarrow R^{\prime}\) be a ring homomorphism. Let \(S\) be a subring of \(R,\) and let \(\tau: S \rightarrow R^{\prime}\) be the restriction of \(\rho\) to \(S\). Show that \(\tau\) is a ring homomorphism and that \(\operatorname{Ker} \tau=\operatorname{Ker} \rho \cap S\)

Suppose \(R_{1}, \ldots, R_{k}\) are rings. Show that for each \(i=1, \ldots, k\), the projection map \(\pi_{i}: R_{1} \times \cdots \times R_{k} \rightarrow R_{i}\) that sends \(\left(a_{1}, \ldots, a_{k}\right)\) to \(a_{i}\) is a surjective ring homomorphism.

Show that if \(R=R_{1} \times R_{2}\) for rings \(R_{1}\) and \(R_{2},\) and \(I_{1}\) is an ideal of \(R_{1}\) and \(I_{2}\) is an ideal of \(R_{2},\) then we have a ring isomorphism \(R /\left(I_{1} \times I_{2}\right) \cong\) \(R_{1} / I_{1} \times R_{2} / I_{2}\)

Let \(\rho: R \rightarrow R^{\prime}\) be a ring homomorphism with kernel \(K\). Let \(I\) be an ideal of \(R\). Show that we have a ring isomorphism \(R /(I+K) \cong \rho(R) / \rho(I)\).

Let \(n\) be a positive integer, and consider the natural map that sends \(a \in \mathbb{Z}\) to \(\bar{a}:=[a]_{n} \in \mathbb{Z}_{n},\) which we may extend coefficient-wise to a ring homomorphism from \(\mathbb{Z}[X]\) to \(\mathbb{Z}_{n}[X],\) as in Example \(7.47 .\) Show that for every \(f \in \mathbb{Z}[X],\) we have a ring isomorphism \(\mathbb{Z}[X] /(f, n) \cong \mathbb{Z}_{n}[X] /(\bar{f})\).