# 6E: Exercises

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1. Prove or disprove: Every finite integral domain is a field.

2. Prove or disprove: Every division ring is an integral domain.

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

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

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

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

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

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

1. Find all idempotents in $$\mathbb{Z}_{12}.$$

2. Find all the idempotents in an integral domain $$R.$$

3. If $$a^m=a^{m+n}, m, n \in \mathbb{N}, \( show that some power of \(a$$ is an idempotent.

4. If $$R$$ is a finite ring, show that some power of each element of $$R$$ is an idempotent.

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

1. Is well defined

2. Is associative

3. Satisfies the distributive laws.

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

1. Find all nilpotents in $$\mathbb{Z}_{12}.$$

2. Find all the nilpotents in an integral domain $$R.$$

3. 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.

This page titled 6E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.