6E: Exercises

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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.