Below, all rings are associative and unital; and the word "ideal" always refers to a two-sided ideal.
Let's stipulate that a ring $R$ has property (P) if every non-unit of $R$ is contained in a completely prime${}^{(1)}$ ideal. It is well known that, under the axioms of ZFC (say), every commutative ring has property (P):In the commutative setting, every non-unit is in fact contained in a maximal ideal, every maximal ideal is a prime ideal, and every prime ideal is completely prime. There are, on the other hand, entire classes of non-commutative rings in everyday life where the same holds true; for instance, local rings and duo ring${}^{(2)}$ have property (P). This leads to the following:
Questions. (i) Is there any special name for a ring with property (P)? (ii) Is there any (interesting) class $\mathcal C$ of non-commutative rings with property (P) that happens to be closed under polynomial extensions, by which I mean that, if $R$ is a ring in the class $\mathcal C$, then so also is the ring of univariate polynomials over $R$?
Unfortunately, neither the class of local rings nor the class of duo rings is closed under polynomial extensions (in the above sense). Note also that a ring $R$ has property (P) only if it is Dedekind-finite (that is, every left- or right-invertible element is a unit).
Notes. (1) An ideal $\mathfrak p$ of a ring $R$ is completely prime if it is proper (in the sense that $\mathfrak p \subsetneq R$) and $ab \in \mathfrak p$ for some $a, b \in R$ implies $a \in \mathfrak p$ or $b \in \mathfrak p$; and is prime if it is proper and $aRb \subseteq \mathfrak p$ for some $a, b \in R$ implies $a \in \mathfrak p$ or $b \in \mathfrak p$ (cf. the article on prime ideals on Wiki.en). (2) A ring $R$ is duo if $aR = Ra$ for every $a \in R$.
