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A group has lots of interesting characteristic subgroups (center, commutator and derived subgroups, Frattini subgroup, Fitting subgroup, the identity component for topological groups, the torsion subgroup for abelian groups). Often, these subgroups play an important role in the classification of the groups.

But what about characteristic subrings? What are interesting examples? I am especially interested in commutative rings, so that the center doesn't really qualify. The only example I know at the moment is the prime ring (the subring generated by the unit).

Since characteristic subgroups are normal and normal subgroups in group theory correspond to ideals in ring theory, perhaps a better question would be: What are interesting characteristic ideals of a (commutative) ring? Here only the nil radical and the Jacobson radical come into my mind.

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  • $\begingroup$ More interesting may be the collection of all prime ideals (the spectrum of the ring). $\endgroup$ – Hagen von Eitzen Oct 24 '12 at 16:43
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The Jacobson radical and nilradical are certainly important, as you mentioned, and I'll throw in a few more. There's the singular ideal and a lot of other radicals like the Artinian radical, von Neumann radical.

Also something conspicuously unmentioned is the socle of the ring.

The singular ideal shows up when studying noncommutative localization, and the Artinian radical is used for the structure of noncommutative noetherian rings. Unfortunately I'm not famliar enough with the results to know what they say about the commutative case.

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  • $\begingroup$ Thank you. It seems to me that almost all these radicals coincide with the nil- or the jacobson radical in the commutative case. $\endgroup$ – Martin Brandenburg Oct 25 '12 at 8:16
  • $\begingroup$ Yes, some of the radicals in the link that I did not mention coincide. The singular ideal and the socle of the ring certainly can be different from the nil and the Jacobson radical, so you probably want to look again at those. I have never thought much about the Artinian and von Neumann radicals, but I will look into it. $\endgroup$ – rschwieb Oct 25 '12 at 12:33
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As far as subrings are concerned, you've already mentioned the prime subring. This is not just a characteristic subring, it's invariant under every ring endomorphism. Wikipedia tells me that I should probably call such a subring a "fully invariant subring."

We can construct another fully invariant subring from the prime subring in the following way. If $R$ is commutative and $S \subseteq R$ is a fully invariant subring, then one readily sees that the integral closure $\overline{S}$ is an integral subring. In particular, the integral closure of the prime ring is a fully invariant subring $\overline{\mathbb{Z} \cdot 1} \subseteq R$.

Regarding ideals, of course the commutator ideal would be a fully invariant ideal in a noncommutative ring.

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