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Problem 18 (Kernels Are Closed Under Multiplication By Arbitrary Elements)

Let $\phi:R\to S$ be ring homomorphism with kernel $K$. Show the following:

  1. The kernel $K$ is a subring of $R$.
  2. If $r\in R$ and $k\in K$, then we have $rk\in K$ and $kr\in K$.

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