Abstract Algebra: Theory and Applications

Find all homomorphisms $\varphi :\mathbb{Z}/6\mathbb{Z}\to \mathbb{Z}/15\mathbb{Z}\text{.}$

Prove that $\mathbb{R}$ is not isomorphic to $\mathbb{C}\text{.}$

Prove or disprove: The ring $\mathbb{Q}(\sqrt{2})=\{a+b\sqrt{2}:a,b\in \mathbb{Q}\}$ is isomorphic to the ring $\mathbb{Q}(\sqrt{3})=\{a+b\sqrt{3}:a,b\in \mathbb{Q}\}\text{.}$

Prove that the Gaussian integers, $\mathbb{Z}[i]\text{,}$ 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.

Use the method of parallel computation outlined in the text to calculate $2234+4121$ by dividing the calculation into four separate additions modulo $95\text{,}$ $97\text{,}$ $98\text{,}$ and $99\text{.}$

Explain why the method of parallel computation outlined in the text fails for $2134\cdot 1531$ if we attempt to break the calculation down into two smaller calculations modulo $98$ and $99\text{.}$

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 $(-1)a=-a\text{.}$

Prove that the associative law for multiplication and the distributive laws hold in $R/I\text{.}$

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

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.

Let $\{I_{\alpha }{\}}_{\alpha \in A}$ be a collection of ideals in a ring $R\text{.}$ Prove that $\bigcap _{\alpha \in A}{I}_{\alpha }$ is also an ideal in $R\text{.}$ Give an example to show that if $I_1$ and $I_2$ are ideals in $R\text{,}$ 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 $n\text{.}$ Show that the set of all nilpotent elements forms an ideal in $R\text{.}$

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.

Let $R$ be a ring, where $a^3=a$ for all $a\in R\text{.}$ 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 $1_S\text{.}$ Prove or disprove that $1_R=1_S\text{.}$

If we do not require the identity of a ring to be distinct from 0, we will not have a very interesting mathematical structure. Let $R$ be a ring such that $1=0\text{.}$ Prove that $R=\{0\}\text{.}$

Prove or disprove: Every finite integral domain is isomorphic to ${\mathbb{Z}}_p\text{.}$

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.

Let $gcd(a,n)=d$ and $gcd(b,d)\ne 1\text{.}$ Prove that $ax\equiv b(\mathrm{mod}n)$ does not have a solution.