= Solution
An <ideal> is an additive subgroup $I\subseteq R$ with $rI\subseteq I$ for all $r\in R$. A <ring homomorphism> preserves addition and multiplication; for the <rings> with identity here use the standard unital convention $\theta(1_R)=1_S$. If $a,b\in\ker\theta$, then $\theta(a-b)=0$, and if $r\in R$ then $\theta(ra)=\theta(r)\theta(a)=0$. Thus the kernel is an <ideal>.
A <maximal ideal> is a proper <ideal> $I\ne R$ such that an <ideal> $J$ satisfying $I\subseteq J\subseteq R$ is either $I$ or $R$.
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