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Problem 36 (The Ideal Test)
Suppose that $A$ is a nonempty subset $A$ of a ring $R$. Prove that $A$ is an ideal of $R$ if
- $a-b\in A$ whenever $a,b\in A$, and
- $ra$ and $ar$ are in $A$ whenever $a\in A$ and $r\in R$.
In other words, prove that an ideal is a nonempty subset that is closed under subtraction and multiplication by an arbitrary ring element.
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