With $\Bbb K$ a commutative rings there a way to characterize $A,B\in\Bbb K^{n\times n}$ with $$Per(AB)=Per(A)Per(B)?$$


John provides a reasonable concept coverage multiplicative property.


How about if I seek $Per(A+B)=Per(A)+Per(B)$ instead of $Per(AB)=Per(A)Per(B)$?



On additive/multiplicative property of permanent
Searching the literature brings up a concept known as permanent groups. A *permanent group* is a group of nonsingular matrices on which the permanent is multiplicative. In was conjectured by M. Marcus in [Permanents](https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/MarvinMarcusHenrykMinc.pdf) that the set nonsingular matrices of the form $PD$, where $P$ is a permutation matrix and $D$ is a diagonal matrix, is a maximal permanent group (see Conjecture 12). This conjecture is proven over the complex numbers by Beasley in [Maximal groups on which the permanent is multiplicative](http://cms.math.ca/10.4153/CJM-1970-024-1). Further work has been done by Beasley and Cummings in the papers titled Permanent Groups, Permanent Groups II, and Permanent Semigroups, etc.


