= Solution
A \b[<unit in a ring>] is an element $u$ with a multiplicative inverse in $R$, so $uv=1$ for some $v\in R$.
An \b[<irreducible element>] is a nonzero nonunit $a$ such that every factorization $a=bc$ has at least one of $b,c$ a <unit in a ring>.
A \b[<prime element>] is a nonzero nonunit $p$ such that $p\mid bc$ implies $p\mid b$ or $p\mid c$. Equivalently its <principal ideal> $(p)$ is a nonzero <prime ideal>. These are definitions in the given commutative <ring>; the equivalence between irreducibility and primality requires additional hypotheses on the <ring>.
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