Obtaining A New Ideal By Adding One Element

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Problem 29 (Obtaining A New Ideal By Adding One Element)

Let $A$ be an ideal of the commutative ring $R$ with unity. We'd like to increase the size of $A$ by adding in a single element $b$. Show the following:

1. Pick $b\in R$. The set $B=\{bc+a\mid c\in R,a\in A\}$ is an ideal of $R$ that contains both the element $b$ and subset $A$.
2. We know $1\in A$ if and only if $A=R$.

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