Let $H$ be a multiplicatively written monoid with identity $1_H$. We write $H^\times$ for the *set of units* (or *invertible elements*) of $H$. We say that an element $a \in H$ is an *atom* if $a \notin H^\times$ and there do not exist $x, y \in H \setminus H^\times$ such that $a = xy$, and a *prime* if $a \notin H^\times$ and $a \mid_H xy$ implies $a \mid_H x$ or $a \mid_H y$. Here, $\mid_H$ is the divisibility preorder on $H$ (that is, $x \mid_H y$ iff $y \in HxH$).

Q.What is known about the class, $\mathcal M_{\sf p}$, of monoids for which every prime is an atom? Have they ever been studied? Do they have a special name?

It is seen that $\mathcal M_{\sf p}$ includes all commutative, unit-cancellative monoids ($H$ is unit-cancellative if $xy = x$ or $yx=x$ for some $x, y \in H$ implies $y \in H^\times$), while a free monoid with basis a set containing at least two elements is a non-commutative, cancellative example.

As for a non-example, it is enough to consider the case when $H$ is a non-trivial monoid with an absorbing element $0_H$, but no zero divisors (e.g., the multiplicative monoid of a domain): Here, $0_H = 0_H \cdot 0_H$ and $0_H \notin H^\times$, so $0_H$ is neither a unit nor an atom. However, $0_H$ is a prime, because $0_H \mid_H xy$, for some $x, y \in H$, only if $x = 0_H$ or $y = 0_H$.