TITLE: Product of a non-unit with any other element
QUESTION [7 upvotes]: In any ring, is the product of a non-unit with any other element necessarily a non-unit?

REPLY [3 votes]: Hint $\rm\ U\,$ closed under divisors $\,\Rightarrow\,$ its complement $\rm\:U'$ closed under multiples.
Proof $\ $ If not, there is $\rm\:n\in U',\:$ with $\rm\:nx\in U,\:$ contra $\rm\,U\,$ closed under divisors.
Note $\ $ The set of units $\rm\,U\,$ is closed under divisors, i.e. if $\rm\,u\,$ is a unit then so is every divisor of $\rm\,u.\,$ Indeed, units are precisely the divisors of $\,1,\:$ hence by transitivity of 'divides' we deduce
$$\rm xy\ unit\ \Rightarrow\ xy\ |\ 1\ \Rightarrow\  x\ |\ xy\ |\ 1\ \Rightarrow\ x\ unit\ $$  
i.e. the set of all divisors of a fixed element is closed under taking divisors (by transitivity).