5E: Excercises

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Apr 9, 2024
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- 5.3: Ring Homomorphisms
- Mock Exams (Celebration of Learning)

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Exercise $\PageIndex{1}$

Prove or disprove: Every finite integral domain is a field.
Prove or disprove: Every division ring is an integral domain.

Exercise $\PageIndex{2}$

List or characterize all of the units in each of the following rings.

Exercise $\PageIndex{3}$

Prove or disprove: $\mathbb{Z}[\sqrt{5}]=\{a+b\sqrt{5}i : a,b \in \mathbb{Z} \}$ is an integral domain.

Exercise $\PageIndex{4}$

$p$ be a prime. Prove that $\mathbb{Z}_{(p)}=\{a/b| a,b \in \mathbb{Z} \, and \, \gcd(b,p)=1 \}$ is a ring.

Exercise $\PageIndex{5}$

Let $R$ be a ring. If $ab+ba=1$ and $a^3=a$ , $\forall a,b\in R,$ then show that $a^2=1.$

Exercise $\PageIndex{6}$

Show that $\mathbb{Q}(\sqrt{5}i) = \{r+s \sqrt{5}i : r,s \in \mathbb{Q} \}$ is a subfield of $\mathbb{C}.$

Exercise $\PageIndex{7}$

An element $e$ of a ring $R$ is said to be idempotent if $e^2=e.$

Find all idempotents in $\mathbb{Z}_{12}.$
Find all the idempotents in an integral domain $R.$
If $a^m=a^{m+n}, m, n \in \mathbb{N},$ show that some power of $a$ is an idempotent.
If $R$ is a finite ring, show that some power of each element of $R$ is an idempotent.

Exercise $\PageIndex{8}$

Show that the multiplication defined for the field of quotients of an integral domain:

Is well defined
Is associative
Satisfies the distributive laws.

Exercise $\PageIndex{9}$

An element $a$ of a ring $R$ is said to be nilpotent if $a^n=0,$ for some $n \in \mathbb{N}.$

Find all nilpotents in $\mathbb{Z}_{12}.$
Find all the nilpotents in an integral domain $R.$

Exercise $\PageIndex{10}$

Let $a, u \in R.$ If $u$ is a unit and $a$ is nilpotent such that $ua=au,$ show that $u+a$ is a unit.

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Exercise 5E.1\PageIndex{1}

Exercise 5E.2\PageIndex{2}

Exercise 5E.3\PageIndex{3}

Exercise 5E.4\PageIndex{4}

Exercise 5E.5\PageIndex{5}

Exercise 5E.6\PageIndex{6}

Exercise 5E.7\PageIndex{7}

Exercise 5E.8\PageIndex{8}

Exercise 5E.9\PageIndex{9}

Exercise 5E.10\PageIndex{10}

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