Chapter 16

Which of the following sets are rings with respect to the usual operations of addition and multiplication? If the set is a ring, is it also a field? (a) \(7 \mathbb{Z}\) (b) \(\mathbb{Z}_{18}\) (c) \(\mathbb{Q}(\sqrt{2})=\\{a+b \sqrt{2}: a, b \in \mathbb{Q}\\}\) (d) \(\mathbb{Q}(\sqrt{2}, \sqrt{3})=\\{a+b \sqrt{2}+c \sqrt{3}+d \sqrt{6}: a, b, c, d \in \mathbb{Q}\\}\) (e) \(\mathbb{Z}[\sqrt{3}]=\\{a+b \sqrt{3}: a, b \in \mathbb{Z}\\}\) (f) \(R=\\{a+b \sqrt[3]{3}: a, b \in \mathbb{Q}\\}\) (g) \(\mathbb{Z}[i]=\left\\{a+b i: a, b \in \mathbb{Z}\right.\) and \(\left.i^{2}=-1\right\\}\) (h) \(\mathbb{Q}(\sqrt[3]{3})=\\{a+b \sqrt[3]{3}+c \sqrt[3]{9}: a, b, c \in \mathbb{Q}\\}\)

Let \(R\) be the ring of \(2 \times 2\) matrices of the form $$ \left(\begin{array}{ll} a & b \\ 0 & 0 \end{array}\right) $$ where \(a, b \in \mathbb{R}\). Show that although \(R\) is a ring that has no identity, we can find a subring \(S\) of \(R\) with an identity.

List or characterize all of the units in each of the following rings. (a) \(\mathbb{Z}_{10}\) (b) \(\mathbb{Z}_{12}\) (c) \(\mathbb{Z}_{7}\) (d) \(\mathbb{M}_{2}(\mathbb{Z}),\) the \(2 \times 2\) matrices with entries in \(\mathbb{Z}\) (e) \(\mathbb{M}_{2}\left(\mathbb{Z}_{2}\right),\) the \(2 \times 2\) matrices with entries in \(\mathbb{Z}_{2}\)

Find all of the ideals in each of the following rings. Which of these ideals are maximal and which are prime? (a) \(\mathbb{Z}_{18}\) (b) \(\mathbb{Z}_{25}\) (c) \(\mathbb{M}_{2}(\mathbb{R}),\) the \(2 \times 2\) matrices with entries in \(\mathbb{R}\) (d) \(\mathbb{M}_{2}(\mathbb{Z}),\) the \(2 \times 2\) matrices with entries in \(\mathbb{Z}\) (e) \(\mathbb{Q}\)

For each of the following rings \(R\) with ideal \(I,\) give an addition table and a multiplication table for \(R / I\). (a) \(R=\mathbb{Z}\) and \(I=6 \mathbb{Z}\) (b) \(R=\mathbb{Z}_{12}\) and \(I=\\{0,3,6,9\\}\)

Find all homomorphisms \(\phi: \mathbb{Z} / 6 \mathbb{Z} \rightarrow \mathbb{Z} / 15 \mathbb{Z}\).

Prove that \(\mathbb{R}\) is not isomorphic to \(\mathbb{C}\).

What is the characteristic of the field formed by the set of matrices $$ F=\left\\{\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 1 \\ 1 & 1 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right)\right\\} $$ with entries in \(\mathbb{Z}_{2} ?\)

Define a map \(\phi: \mathbb{C} \rightarrow \mathbb{M}_{2}(\mathbb{R})\) by $$ \phi(a+b i)=\left(\begin{array}{cc} a & b \\ -b & a \end{array}\right) $$ Show that \(\phi\) is an isomorphism of \(\mathbb{C}\) with its image in \(\mathbb{M}_{2}(\mathbb{R})\).

Prove that the Gaussian integers, \(\mathbb{Z}[i]\), are an integral domain.

Prove that \(\mathbb{Z}[\sqrt{3} i]=\\{a+b \sqrt{3} i: a, b \in \mathbb{Z}\\}\) is an integral domain.

Solve each of the following systems of congruences. $$(a) \begin{array}{ll} x \equiv 2 & (\bmod 5) \\ x \equiv 6 & (\bmod 11 \end{array} $$ $$(b) \begin{array}{ll} x \equiv 3 & (\bmod 7) \\ x \equiv 0 & (\bmod 8) \\ x \equiv 5 & (\bmod 15 \end{array} $$ $$(c) \begin{aligned} &x \equiv 2 \quad(\bmod 4)\\\ &x \equiv 4 \quad(\bmod 7)\\\ &x \equiv 7 \quad(\bmod 9)\\\ &x \equiv 5 \quad(\bmod \end{aligned} $$ $$(d) \begin{array}{ll} x \equiv 3 & (\bmod 5) \\ x \equiv 0 & (\bmod 8) \\ x \equiv 1 & (\bmod 11) \\ x \equiv 5 & (\bmod 13) \end{array} $$

If \(R\) is a field, show that the only two ideals of \(R\) are \(\\{0\\}\) and \(R\) itself.

Let \(a\) be any element in a ring \(R\) with identity. Show that \((-1) a=-a .\)

Let \(\phi: R \rightarrow S\) be a ring homomorphism. Prove each of the following statements. (a) If \(R\) is a commutative ring, then \(\phi(R)\) is a commutative ring. (b) \(\phi(0)=0\). (c) Let \(1_{R}\) and \(1_{S}\) be the identities for \(R\) and \(S\), respectively. If \(\phi\) is onto, then \(\phi\left(1_{R}\right)=1_{S}\). (d) If \(R\) is a field and \(\phi(R) \neq 0,\) then \(\phi(R)\) is a field.

Prove that the associative law for multiplication and the distributive laws hold in \(R / I\)

Prove the Second Isomorphism Theorem for rings: Let \(I\) be a subring of a ring \(R\) and \(J\) an ideal in \(R\). Then \(I \cap J\) is an ideal in \(I\) and $$I / I \cap J \cong I+J / J$$

Prove the Third Isomorphism Theorem for rings: Let \(R\) be a ring and \(I\) and \(J\) be ideals of \(R\), where \(J \subset I\). Then $$R / I \cong \frac{R / J}{I / J}$$

Prove the Correspondence Theorem: Let \(I\) be an ideal of a ring \(R\). Then \(S \rightarrow S / I\) is a one-to-one correspondence between the set of subrings \(S\) containing \(I\) and the set of subrings of \(R / I\). Furthermore, the ideals of \(R\) correspond to ideals of \(R / I\).

Let \(R\) be a ring and \(S\) a subset of \(R\). Show that \(S\) is a subring of \(R\) if and only if each of the following conditions is satisfied. (a) \(S \neq \emptyset\). (b) \(r s \in S\) for all \(r, s \in S\). (c) \(r-s \in S\) for all \(r, s \in S\).

Let \(R\) be a ring with a collection of subrings \(\left\\{R_{\alpha}\right\\}\). Prove that \(\bigcap R_{\alpha}\) is a subring of \(R\). Give an example to show that the union of two subrings is not necessarily a subring.

Let \(\left\\{I_{\alpha}\right\\}_{\alpha \in A}\) be a collection of ideals in a ring \(R\). Prove that \(\bigcap_{\alpha \in A} I_{\alpha}\) is also an ideal in \(R\). Give an example to show that if \(I_{1}\) and \(I_{2}\) are ideals in \(R\), then \(I_{1} \cup I_{2}\) may not be an ideal.

Let \(R\) be an integral domain. Show that if the only ideals in \(R\) are \\{0\\} and \(R\) itself, \(R\) must be a field.

Let \(R\) be a commutative ring. An element \(a\) in \(R\) is nilpotent if \(a^{n}=0\) for some positive integer \(n\). Show that the set of all nilpotent elements forms an ideal in \(R\).

A ring \(R\) is a Boolean ring if for every \(a \in R, a^{2}=a\). Show that every Boolean ring is a commutative ring.

Let \(R\) be a ring, where \(a^{3}=a\) for all \(a \in R\). Prove that \(R\) must be a commutative ring.

Let \(R\) be a ring with identity \(1_{R}\) and \(S\) a subring of \(R\) with identity \(1_{S}\). Prove or disprove that \(1_{R}=1_{S}\)

Let \(S\) be a nonempty subset of a ring \(R\). Prove that there is a subring \(R^{\prime}\) of \(R\) that contains \(S\).

Let \(R\) be a ring. Define the center of \(R\) to be $$ Z(R)=\\{a \in R: a r=r a \text { for all } r \in R\\} $$ Prove that \(Z(R)\) is a commutative subring of \(R\).

Let \(p\) be prime. Prove that $$ \mathbb{Z}_{(p)}=\\{a / b: a, b \in \mathbb{Z} \text { and } \operatorname{gcd}(b, p)=1\\} $$ is a ring. The ring \(\mathbb{Z}_{(p)}\) is called the ring of integers localized at \(p\).

Prove or disprove: Every finite integral domain is isomorphic to \(\mathbb{Z}_{p}\).

Let \(R\) be a ring with identity. (a) Let \(u\) be a unit in \(R\). Define a map \(i_{u}: R \rightarrow R\) by \(r \mapsto u r u^{-1}\). Prove that \(i_{u}\) is an automorphism of \(R\). Such an automorphism of \(R\) is called an inner automorphism of \(R\). Denote the set of all inner automorphisms of \(R\) by \(\operatorname{Inn}(R)\). (b) Denote the set of all automorphisms of \(R\) by Aut \((R)\). Prove that \(\operatorname{Inn}(R)\) is a normal subgroup of \(\operatorname{Aut}(R)\) (c) Let \(U(R)\) be the group of units in \(R\). Prove that the map $$ \phi: U(R) \rightarrow \operatorname{Inn}(R) $$ defined by \(u \mapsto i_{u}\) is a homomorphism. Determine the kernel of \(\phi\).

Let \(R\) and \(S\) be arbitrary rings. Show that their Cartesian product is a ring if we define addition and multiplication in \(R \times S\) by (a) \((r, s)+\left(r^{\prime}, s^{\prime}\right)=\left(r+r^{\prime}, s+s^{\prime}\right)\) (b) \((r, s)\left(r^{\prime}, s^{\prime}\right)=\left(r r^{\prime}, s s^{\prime}\right)\)

An element \(x\) in a ring is called an idempotent if \(x^{2}=x\). Prove that the only idempotents in an integral domain are 0 and 1 . Find a ring with a idempotent \(x\) not equal to 0 or 1

Let \(\operatorname{gcd}(a, n)=d\) and \(\operatorname{gcd}(b, d) \neq 1\). Prove that \(a x \equiv b(\bmod n)\) does not have a solution.

The Chinese Remainder Theorem for Rings. \(\quad\) Let \(R\) be a ring and \(I\) and \(J\) be ideals in \(R\) such that \(I+J=R\). (a) Show that for any \(r\) and \(s\) in \(R,\) the system of equations $$ \begin{array}{ll} x \equiv r & (\bmod I) \\ x \equiv s & (\bmod J) \end{array} $$ has a solution. (b) In addition, prove that any two solutions of the system are congruent modulo \(I \cap J\). (c) Let \(I\) and \(J\) be ideals in a ring \(R\) such that \(I+J=R\). Show that there exists a ring isomorphism $$ R /(I \cap J) \cong R / I \times R / J $$