Units and ideal

Question

Suppose $A$ is principal ideal. For $x\in A$

Prove that $$ x \ \ is \ \ unit \longleftrightarrow (x)=A=(1)$$ I don't know where to start. Please help.

Answer 1

Hints:

• An element $x\in A$ is a unit by definition means that there is some element $y\in A$ such that $xy=1_A$.
• The ideal $(x)$ equals $A$ if and only if it contains every element of $A$, in particular it contains $1_A$.
• $(x)=\{xy\mid y\in A\}$.

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Hope this helps.

Answer 2

$\rightarrow$

Suppose that x is a unit. So by definition $xy=1_A$ for some $y\in A.$

Now $(x)=\{xy\mid y\in A\} =1_A=(1)$ $\ \ \rightarrow (x)=A=(1)$

$\leftarrow$

if we suppose that $(x)=A=(1)$ then $(x)=\{xy\mid y\in A\} =1_A=(1)$ $\ \ \ \ \rightarrow xy=1_A$ so $x$ is unit.