# Is the characteristic of a ring related to characteristic subgroups of the additive group?

Recently, I was looking into properties of ring characteristics, and I stumbled upon the definition of a "characteristic subgroup," which despite using the same name doesn't seem to me to be obviously related. However, since every ring induces an additive group, I was wondering if there was a deeper connection between these two concepts, or whether the common name is just a coincidence. Or what about for field characteristics and characteristic subgroups of the multiplicative group?

Based on a very cursory Google search I wasn't able to find a connection, just separate discussions for similarly named concepts.

• There is no relation between the two things. There are tons of characteristic blahs on math... Feb 10, 2023 at 9:40

One way to define the characteristic of a ring is to look at the kernel of the unique ring morphism $$u:\mathbb{Z}\to R$$. It is starightforward to observe that $$\ker u=(\operatorname{char} R)$$. So we might ask if $$\mathbb{Z}/(\operatorname{char} R)\hookrightarrow R$$ is a characteristic subgroup of $$(R,+)$$?
This is not the case, not even when $$R$$ is a field. For example, if $$R=\mathbb{F}_{p^2}$$, then the its characteristic is $$p$$ and we obtain an inclusion $$\mathbb{F}_{p}\hookrightarrow\mathbb{F}_{p^2}$$. This makes $$\mathbb{F}_{p^2}$$ a two-dimensional $$\mathbb{F}_{p}$$-vector space, so in particular $$(\mathbb{F}_{p^2},+)\cong(\mathbb{F}_{p}\times\mathbb{F}_{p},+)$$. Furthermore, the vector space automorphisms $$\operatorname{GL}(2,\mathbb{F}_{p})$$ certainly don't fix any one-dimensional subspace simultaneously, so $$(\mathbb{F}_{p},+)\hookrightarrow(\mathbb{F}_{p^2},+)$$ isn't characteristic.
What somewhat resembles the property of characteristic subgroups in this context, is that if $$\sigma:R\to R$$ is any ring automorphism, then $$\sigma$$ preserves $$\mathbb{Z}/(\operatorname{char} R)\hookrightarrow R$$. So you might think of it as a characteristic subring for ring automorphisms.