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  3. let d and d'be integral domains. let c = chard and

Let D and D'be integral domains. Let c = charD and c'= charD' (a) Prove that the direct...

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Advance Math

LetD and D'be integral domains.
Let c = charD and c'= charD'

(a) Prove that the direct product D ×D'has unity.

(b) Let a ∈D and b∈D'.
Prove that (a, b) is a unit in D ×D'⇐⇒
a is a unit in D, b is a unit in D'.

(c) Prove that D×D'is never an integral domain.

(d) Prove that if c, c'> 0,
then char(D ×D') = lcm(c, c')

(e) Prove that if c = 0, then char(D ×D') = 0.

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