Chapter 2.5, Problem 1E

Let $\left(R,+,\cdot \right)$ be an algebraic structure such that

1. $\left(R,+\right)$ is an abelian group with identity 0.
2. $\left(R-\left\{0\right\},\cdot \right)$ is an abelian group with identity 1.
3. For all a, b, $c\in R,a\cdot \left(b+c\right)=\left(a\cdot b\right)+\left(a\cdot c\right).$
4. $0\ne 1.$

Prove that $\left(R,+,\cdot \right)$ is a field by showing that

1. for all a, b, $c\in R,\left(a+b\right)\cdot c=\left(a\cdot c\right)+\left(b\cdot c\right).$
2. R has no divisors of zero.