91Ó°ÊÓ

Find all possible ring homomorphisms between the indicated rings. $$ \phi: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z} $$

Show that \(\mathbb{Z} \times\\{0\\}\) is an ideal in \(\mathbb{Z} \times \mathbb{Z}\).

In Exercises 1 through 15 determine the field of quotients of the indicated rings if it exists. If it does not exist, explain why. $$ \mathbb{Z}_{12} /\langle 4\rangle $$

Show that \(\mathbb{Z}_{n} \times \mathbb{Z}_{m}\) and \(\mathbb{Z}_{n m}\) are isomorphic rings if and only if \(n\) and \(m\) are relatively prime.

In Exercises 1 through 15 determine the field of quotients of the indicated rings if it exists. If it does not exist, explain why. $$ \mathbb{Z}_{15} /\langle 3\rangle $$

For \(a, b \in \mathbb{Z},\) let \(B(a, b) \in M(2, \mathbb{Z})\) be defined by \(B(a, b)=\left[\begin{array}{cc}a & 3 b \\ b & a\end{array}\right]\). Let \(S=\) \(\\{B(a, b) \mid a, b \in \mathbb{Z}\\} \subseteq M(2, \mathbb{Z}) .\) Show that \(S \propto \mathbb{Z}[\sqrt{3}]=\\{a+b \sqrt{3} \mid a, b \in \mathbb{Z}\\}\).

In Exercises 1 through 15 determine the field of quotients of the indicated rings if it exists. If it does not exist, explain why. $$ Q(t)=\left\\{a+b i \mid a, b \in Q, i^{2}=-1\right\\} $$

Show that every subfield of \(\mathbb{R}\) must contain \(\mathbb{Q}\).

Show that \(\mathbb{R}\) and \(\mathbb{C}\) are not isomorphic rings.

Let \(I\) and \(J\) be ideals in a ring \(R\). Show that \(I \cap J\) is an ideal in \(R\).