Ideals and Quotient Rings

Let $I$ be an ideal of a ring $R$. Show that $I$ is a subring of $R$.

Let the sets

It is given that $R$ is a subring of $\mathcal{M}_{2\times 2}(\mathbb{R})$.

1. Show that $I$ is a subring of $R$.

2. Show that $I$ is an ideal of $R$.

3. Simplify

   in $R/I$.

Let $R$ be a ring with identity 1, and let $I$ be an ideal of $R$.

1. If $1\in I$, what does this imply about $I$?

2. Prove $I$ contains a unit if and only if $I=R$.

Let $R$ be a ring with ideals $I$ and $J$. Prove that $I\cap J$ is an ideal of $R$.

Show that all principal ideals are ideals.

Show that the following ideals are principal in any commutative ring with identity.

1. The trivial ideal.

2. The ring itself.

Is the principal ideal $\left\langle 2\right\rangle=\{0,2,4,6\}$ prime in $\mathbb{Z}_{8}$?

What condition(s) must be placed on the positive integer $n$ so that $n\mathbb{Z}$ is a maximal ideal in $\mathbb{Z}$?

Is $\left\langle 3-i\right\rangle$ a prime ideal in the Gaussian integers?

Let $R$ be a finite commutative ring with identity, and let $P$ be a prime ideal in $R$. Show that $P$ is maximal in $R$.

Let $R$ be a commutative ring and let $A$ be a non-empty subset of $R$. Show that $\mathrm{Ann}_{R}(A)$ is an ideal of $R$.

Find $\mathrm{Ann}_{\mathbb{Z}_{36}}(\{15\})$.

Let $S=\{a+2bi\mid a,b\in\mathbb{Z}\}$. Show that $S$ is a subring of $\mathbb{Z}[i]$ but not an ideal of $\mathbb{Z}[i]$.

Consider the subset

of $\mathcal{M}_{2\times 2}(\mathbb{Z})$. Show that $I$ is an ideal of $\mathcal{M}_{2\times 2}(\mathbb{Z})$.

Let $R$ be a commutative ring with identity and let $I$ be an ideal of $R$.

1. Prove that if $R$ is a field then $I$ is either the trivial ring or $R$. That is, prove that $R$ has no proper ideals. Hence prove that any field is a PID.

2. Prove that if $R$ has no proper ideals, then $R$ is a field.

Let $R$ be a commutative ring and let $I$ and $J$ be ideals in $R$. Prove that

1. $\sqrt{\sqrt{I}}=\sqrt{I}$; and

2. $\sqrt{I\cap J}=\sqrt{I}\cap\sqrt{J}$.

Let $m$ and $n$ be positive integers, and let $d=\gcd(m,n)$ and $l=\mathrm{lcm}(m,n)$. Prove that

1. $m\mathbb{Z}\cap n\mathbb{Z}=l\mathbb{Z}$; and

2. $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$.

Let $R$ be a commutative ring. Prove that $R/\mathfrak{N}_{R}$ has no non-zero nilpotents.

Prove that every non-trivial prime ideal is a maximal ideal in a PID.

Prove 1.20.