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MATH21112 Rings and Fields
Example Sheet 3 - Domains
1. Prove that if R is a domain then char(R) = 0 or char(R) is a prime integer.
2. Is Mn (R) a domain?
3. Prove that in a domain R the cancellation rule applies
ie. for all a,b, c ∈ R, if ab = ac and a ≠ 0 then b = c.
Give an example of a ring where the cancellation rule does not hold.
4. Suppose that R is a domain and let a, b ∈ R with a ≠ 0. Prove that the equation ax = b has at most one solution for x in R.
5. Show that any ring with exactly three elements must be a domain.
6. Let p be a prime. Show that for any [a]p, [b]p ∈ Zp
([a]p + [b]p )p = ([a]p )p + ([b]p )p.