The largest algebraic structure in which every irreducible is prime We know that in a UFD every irreducible is prime. 
But this document rise up in my mind a question that; is there a known algebraic structure (larger than a UFD) in which every irreducible is prime?
Is there a name for such integral domains?
Thanks for sharing any knowledge,
 A: A domain is called atomic if each non-zero and  non-invertible elements is a product of irreducible elements. 
It can be shown that for an atomic domain $D$ it is equivalent: 


*

*$D$ is a UFD.

*Each irreducible in $D$ is prime.
That is for atomic domains there is nothing but UFDs. 
On the other hand a UFD is always atomic; as a prime is always irreducible. 
Thus, if one wants to have something beyond UFDs one needs to leave the realm of domains where each element has a factorization into irreducibles.  As Bill Dubuque mentions this is well possible and done. The "largest," that is most inclusive, defintion you seek would be just: "a domain where each irreducible is prime." A name for this is AP-domain (see comment by OP). 
In such a domain it is true that every non-zero non-identity element has at most one factorization into irreducibles (up to multiplication by units and reordering). 
Note that for a UFD one has the characterization  each non-zero non-identity element has exactly one factorization into irreducibles (up to multiplication by units and reordering)
Put differently, in an AP-domain it is true that each element that has a factorization into irreducibles at all has a (essentially) unique factorization. 
Such a domain is called an unrestricted UFD by Coykendall and Zafrullah. Thus, every AP-domain is an unrestricted UFD. The converse is not true though as shown by Coykendall and Zafrullah (in contrast to the situation for atomic domains). 
A relevant paper is: Jim Coykendall, Muhammad Zafrullah, AP-domains and unique factorization, Journal of Pure and Applied Algebra (2004).
A: It so happens that I addressed this question in hd1203, in the year 2012. Those looking for a characterization for domains in which atoms are prime may want to look up: http://www.lohar.com/mithelpdesk/hd1203.pdf
In the above write up, that ended in 2012, you will find descriptions of domains in which atoms are usually prime, domains such as pre-Schreier domains, PSP domains etc.
