Ring Homomorphisms and Isomorphisms

Let the function $\phi:\mathcal{M}_{2\times 2}(\mathbb{Z})\to\mathbb{Z}$ be defined such that

Is $\phi$ a ring homomorphism?

Let $R$ and $S$ be rings with additive identities $0_{R}$ and $0_{S}$ respectively. Show that the trivial homomorphism $\phi:R\to S,r\mapsto 0_{S}$ is, indeed, a ring homomorphism.

Let $R$ be a ring. Show that the identity homomorphism $\mathrm{id}:R\to R,r\mapsto r$ is a ring endomorphism.

Show that the identity homomorphism is actually an automorphism.

Let $R_{1}$ and $R_{2}$ be rings, and $\phi:R_{1}\to R_{2}$ be a ring homomorphism.

1. Show that $\phi(0_{1})=0_{2}$, where $0_{1}$ and $0_{2}$ are the additive identities of $R_{1}$ and $R_{2}$ respectively.

2. If $R_{1}$ and $R_{2}$ are division rings, then show that $\phi(1_{1})=1_{2}$, where $1_{1}$ and $1_{2}$ are the multiplicative identities of $R_{1}$ and $R_{2}$ respectively.

Let $R_{1}$ and $R_{2}$ be rings, $x\in R_{1}$, and $\phi:R_{1}\to R_{2}$ be a ring homomorphism.

1. Show that $\phi(-x)=-\phi(x)$.

2. If $R_{1}$ and $R_{2}$ are division rings, show that $\phi(x^{-1})=(\phi(x))^{-1}$.

Let $R_{1}$ and $R_{2}$ be rings, and let $\phi:R_{1}\to R_{2}$ be a ring homomorphism. Prove that $\ker\phi$ is an ideal of $R_{1}$.

Show that $\mathbb{Z}_{n}\cong\mathbb{Z}/n\mathbb{Z}$ by considering the ring homomorphism $\phi:\mathbb{Z}\to\mathbb{Z}_{n}$ where $m\mapsto m\mod n$ and by using the FRIT (1.23).

For the map $\phi$ in 1.27, show that $\phi(n)=n$ for all positive integers $n$.

Suppose $R$ and $R^{\prime}$ are rings such that $\mathbb{Q}$ is a subring of both $R$ and $R^{\prime}$. Let $\phi:R\to R^{\prime}$ be a ring homomorphism such that $\phi(1)=1$. Show that for any $q\in\mathbb{Q}$ we have $\phi(q)=q$.

Let $R$ be a ring.

1. Show that $R/\{0\}\cong R$.

2. Prove that $R$ is an integral domain if and only if $\{0\}$ is a prime ideal.

3. Prove that $R$ is a field if and only if $\{0\}$ is a maximal ideal.

Find all ring endomorphisms of $\mathbb{Q}$.

Show $\mathbb{Z}^{2}\not\cong\mathbb{Q}$.

Show $\mathbb{Q}[\sqrt{2}]\not\cong\mathbb{Q}[\sqrt{3}]$.

Consider the subring

of $\mathcal{M}_{2\times 2}(\mathbb{Z})$. Show that $R\cong\mathbb{Z}^{2}$.

Let $R$ and $R^{\prime}$ be rings, and let $\phi:R\to R^{\prime}$ be a ring isomorphism. Prove or disprove the following statements.

1. $\phi^{-1}:R^{\prime}\to R$ is a ring isomorphism.

2. If $R$ has a subring with $n$ elements, then so does $R^{\prime}$.

3. If $R$ has an ideal, then so does $R^{\prime}$.

Find all ring endomorphisms of $\mathbb{Z}_{10}$. Hence find all ring automorphisms $\psi$ of $\mathbb{Z}_{10}$.

Find all ring endomorphisms of $\mathbb{Q}[\sqrt{3}]$. Hence find all ring automorphisms $\psi$ of $\mathbb{Q}[\sqrt{3}]$.

Let $R$ and $R^{\prime}$ be commutative rings, $I$ be an ideal of $R$, and $\phi:R\to R^{\prime}$ be a ring homomorphism.

1. Show that $\phi(\sqrt{I})\subseteq\sqrt{\phi(I)}$.

2. If $\phi$ is surjective with $\ker\phi\subseteq I$, prove that $\phi(\sqrt{I})=\sqrt{\phi(I)}$.

Let $R$ be a ring with a subring $S$ and ideal $I$. Prove that

1. $S+I$ is a subring of $R$;

2. $S\cap I$ is an ideal of $S$; and

3. $S/(S\cap I)\cong(S+I)/I$.

Let $R$ be a ring with ideals $I$ and $J$ such that $I$ is a subset of $J$.

1. Prove that $J/I$ is an ideal of $R/I$.

2. Prove that $\frac{R/I}{J/I}\cong R/J$.(Note: remember to prove that the map is well-defined.)