Prove that the Gaussian integers, \({\mathbb Z}[i ]\text{,}\) are an integral domain.
5.
Prove that \({\mathbb Z}[ \sqrt{3}\, i ] = \{ a + b \sqrt{3}\, i : a, b \in {\mathbb Z} \}\) is an integral domain.
6.
Prove that
\begin{equation*}
{\mathbb Z}\left[\textstyle\frac{1+\sqrt{5}}{2}\right] = \left\{a + b \frac{1+\sqrt{5}}{2} : a, b \in \mathbb Z\right\}
\end{equation*}
is a subring of \(\mathbb R\text{.}\)
7.
Let \(S\) be a set and define the following operations on the power set \(\mathcal P(S)\text{,}\) which is defined to be the set of all subsets of \(S\text{:}\)
\(\displaystyle A + B = A \cup B \setminus (A \cap B)\)
\(\displaystyle A \cdot B = A \cap B\)
Show that these operations define a ring on \(\mathcal P(X)\text{.}\) What is the characteristic of \(\mathcal P(X)\text{?}\)
8.
Prove or disprove: Any subring of a field \(F\) is an integral domain.
9.
A ring \(R\) is a Boolean ring if for every \(a \in R\text{,}\)\(a^2 = a\text{.}\) Show that every Boolean ring is a commutative ring.
10.
Let \(R\) be a ring, where \(a^3 =a\) for all \(a \in R\text{.}\) Prove that \(R\) must be a commutative ring.
11.
If the identity of a ring is not distinct from 0, we will not have a very interesting mathematical structure. Let \(R\) be a ring such that \(1 = 0\text{.}\) Prove that \(R = \{ 0 \}\text{.}\)
12.
Let \(R\) be a ring and \(S\) a subset of \(R\text{.}\) Show that \(S\) is a subring of \(R\) if and only if each of the following conditions is satisfied.
\(1 \in S\text{.}\)
\(rs \in S\) for all \(r, s \in S\text{.}\)
\(r - s \in S\) for all \(r, s \in S\text{.}\)
13.
Let \(R\) be a ring with a collection of subrings \(\{ R_{\alpha} \}\text{.}\) Prove that \(\bigcap R_{\alpha}\) is a subring of \(R\text{.}\) Give an example to show that the union of two subrings is not necessarily a subring.
14.
Prove that the only subring of \(\mathbb Z\) is \(\mathbb Z\text{.}\)
15.
Let \(R\) be a ring. Define the center of \(R\) to be
\begin{equation*}
Z(R) = \{ a \in R : ar = ra \text{ for all } r \in R \}.
\end{equation*}
Prove that \(Z(R)\) is a commutative subring of \(R\text{.}\)
16.
Show that if \(R\) is any ring, then there is a unique homomorphism \(\phi \colon \mathbb Z \rightarrow R\text{.}\)
17.
Prove that \({\mathbb R}\) is not isomorphic to \({\mathbb C}\text{.}\)
18.
Prove or disprove: The ring \({\mathbb Q}( \sqrt{2}\, ) = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \}\) is isomorphic to the ring \({\mathbb Q}( \sqrt{3}\, ) = \{a + b \sqrt{3} : a, b \in {\mathbb Q} \}\text{.}\)
19.
Let \(\phi \colon R \rightarrow S\) be a ring homomorphism. Prove each of the following statements.
If \(R\) is a commutative ring, then \(\phi(R)\) is a commutative ring.
\(\phi( 0 ) = 0\text{.}\)
If \(R\) is a field, then \(\phi(R)\) is a field.
20.
Let \(p\) be prime. Prove that
\begin{equation*}
{\mathbb Z}_{(p)} = \{ a / b : a, b \in {\mathbb Z} \text{ and } \gcd( b,p) = 1 \}
\end{equation*}
is a ring. The ring \({\mathbb Z}_{(p)}\) is called the ring of integers localized at \(p\text{.}\)
21.
The following problems concern the ring of quaternions \(\mathbb
H\) as in Example 4.7.
Compute the products \((1+i)(1+i+j)\) and \((1+i+j)(1+i)\text{.}\)
Find the inverse of \(1+k\text{.}\)
22.
Let \(\mathbb H\) be the ring of quaternions from Example 4.7. Let \(x, y, z\) be real numbers such that \(x^2+y^2+z^2=1\) and set \(\alpha = x\mathbf i + y \mathbf j + z \mathbf k\text{.}\)
Show that \(\alpha^2 = -1\text{.}\)
For \(a\in\mathbb R\text{,}\) show that \(\alpha a = a \alpha\text{.}\)
Define \(\phi\colon\mathbb C \rightarrow \mathbb H\) by \(\phi(a+bi)=a+b\alpha\) and show that \(\phi\) is a injective ring homomorphism.
23.
Prove or disprove: Every finite integral domain is isomorphic to \({\mathbb Z}_p\text{.}\)
24.
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
\(\displaystyle (r, s) + (r', s') = ( r + r', s + s')\)
\(\displaystyle (r, s)(r', s') = ( rr', ss')\)
25.
An element \(x\) in a ring is called an idempotent if \(x^2 = x\text{.}\) Prove that the only idempotents in an integral domain are \(0\) and \(1\text{.}\) Find a ring with a idempotent \(x\) not equal to 0 or 1.
26.
Suppose that \(x\) and \(y\) are idempotents in a commutative ring R.
Show that \(1-x\) is an idempotent.
Show that \(xy\) is an idempotent.
Show that \(x + y -xy\) is an idempotent.
27.
Find all idempotents in the ring \(\mathbb Z_{12}\text{.}\)
28.
A field \(F\) is called a prime field if it has no proper subfields. If \(E\) is a subfield of \(F\) and \(E\) is a prime field, then \(E\) is a prime subfield of \(F\text{.}\)
Prove that every field contains a unique prime subfield.
If \(F\) is a field of characteristic 0, prove that the prime subfield of \(F\) is isomorphic to the field of rational numbers, \({\mathbb Q}\text{.}\)
If \(F\) is a field of characteristic \(p\text{,}\) prove that the prime subfield of \(F\) is isomorphic to \({\mathbb Z}_p\text{.}\)
29.
Let \(D\) be an integral domain.
Show that the operation of multiplication is well-defined in the field of fractions, \(F_D\text{.}\)
Verify the associative and commutative properties for addition in \(F_D\text{.}\)
Verify the associative and commutative properties for multiplication in \(F_D\text{.}\)
30.
Let \(R\) be a commutative ring with identity. We define a multiplicative subset of \(R\) to be a subset \(S\) such that \(1 \in S\) and \(ab \in S\) if \(a, b \in S\text{.}\)
Define a relation \(\sim\) on \(R \times S\) by \((a, s) \sim (a', s')\) if there exists an \(s^\ast \in S\) such that \(s^\ast(s' a -s a') = 0\text{.}\) Show that \(\sim\) is an equivalence relation on \(R \times S\text{.}\)
Let \(a/s\) denote the equivalence class of \((a,s) \in R \times S\) and let \(S^{-1}R\) be the set of all equivalence classes with respect to \(\sim\text{.}\) Define the operations of addition and multiplication on \(S^{-1} R\) by
respectively. Prove that these operations are well-defined on \(S^{-1}R\) and that \(S^{-1}R\) is a ring with identity under these operations. The ring \(S^{-1}R\) is called the ring of quotients of \(R\) with respect to \(S\text{.}\)
