Prove that an element of an integral domain is a unit iff it generates the domain.
To prove that an element of an integral domain is a unit if and only if it generates the domain, we'll need to prove two implications:
1. If an element is a unit, then it generates the integral domain.
2. If an element generates the integral domain, then it is a unit.
Let's start with the first implication:
1. If an element is a unit, then it generates the integral domain.
To prove this, let's assume that "a" is a unit in the integral domain "D." This means that there exists an element "b" in "D" such that "ab = 1," where "1" is the multiplicative identity of the integral domain. We want to show that "a" generates the entire domain "D."
Take any element "x" in "D." We need to show that there exists an element "y" in "D" such that "ay = x."
Since "a" is a unit, we know that there exists an
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