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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.

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

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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.

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  • $\begingroup$ Thanks for your answer! $\endgroup$ Feb 11, 2023 at 2:59

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