If $x \in R$ is irreducible then $x u$ and $xy$ are irreducible where $u \in R^*$ is a unit and $y \in R$ is irreducible.
Let $R$ be a ring.
How do I see that if $x \in R$ is irreducible then:
$x u$ is irreducible, where $u \in R^*$ is a unit
$xy$ is irreducible, where $y \in R$ is irreducible
I've been trying to reach a contradiction, but I'd no luck in doing so.
Showing $xu$ is irreducible is a straightforward application of the definition. Proof by contradiction is possible, but usually you pass it by in favor of a direct proof.
Suppose $xu=ab$. Then $x=abu^{-1}$. Since $x$ is irreducible, $a$ is a unit or else $bu^{-1}$ is a unit. If $bu^{-1}=v$ is a unit, then since the product of units is a unit, $b=vu$ is a unit. Thus either $a$ or $b$ is a unit, and we've verified $xu$ is irreducible.
As noted in the comments, the second point is clearly false. If $x,y$ are two irreducibles, then $xy$ is clearly a product of two things which aren't units. You could say that $xy$ is always reducible in such a case. Maybe that was the intent.