I am giving a series of lectures where I introduce some undergraduates to basic ideas from category theory. One of the things I would like to show them is how category theory could be used to make precise some of the intuitive ideas they have been exposed to (e.g., what "natural" means when we speak about the natural isomorphism of a finite dimensional vector space with its double dual, what "universal" really means when we speak of a universal construction, etc.).
One of the subtleties I would like to illustrate with explicit examples is the difference between "property" and "structure". I would really like to show them that "being a preadditive category" (by preadditive here I just mean $Ab$-enriched category, I do not assume existence of a $0$-object) is not a property, so it does not make sense to give me an abstract category and ask me if it is preadditive or not (I can ask whether it can be made into a preadditive category, but even in that case the preadditive structure may not be unique).
To illustrate this, I would like to find some minimal example of two different $Ab$-enrichments of the same abstract category. A good way to do that seems that of finding two (unitary and associative) rings $(R,+,.)$ and $(S,+',*)$, such that $(R,.)\cong (S,*)$ as monoids (i.e., as one-object categories) but $(R,+)\not\cong (S,+')$ as Abelian groups.
Do you have some easy example of two rings with the properties described above?
Best Answer
I think this works. Take $R=\mathbb{Z}[x]$, and $S=\mathbb{F}_3[x]$. They certainly have different additive structure, since $R$ has characteristic $0$ but $S$ has characteristic $3$.
Since they are both UFDs, the multiplicative structure on the nonzero elements will be of the form $U\times M$, where $U$ is the unit group, and $M$ is the free commutative monoid of rank $\kappa$, where $\kappa$ is the number of irreducibles up to associates. The unit group of both $R$ and $S$ is cyclic of order $2$; $\mathbb{Z}[x]$ and $\mathbb{F}_3[x]$ both have $\aleph_0$ non-associate irreducibles. Now adjoin a zero element to both of these to get the full multiplicative structures of $R$ and $S$. So their multiplicative structures are isomorphic.