Similarities Between Rings and Groups: Ideals and Normal Subgroups

Question

I've been pondering the parallels between rings and groups and the commonalities shared by ideals and normal subgroups. There's a theorem stating that if $R$ is a ring and it happens to be a field, then all its ideals are trivial, i.e., $\{0\}$ and $R$ itself.

So, my question is this: Suppose $F$ is a field, and we consider the set $(F, 0_{F}, +)$ solely as a group (without any ring structure). Are all of its normal subgroups trivial subgroups?

I'd appreciate your insights on this matter. Thanks!

Looking forward to your insights and explanations.

In a group only one binary operation is given. But a field has two binary operations. How is another binary operation defined when you say a group has a field structure? — Asigan, Oct 22 '23 at 10:01
so let's start with F that is a field and look upon F as a addition group with it's "+": (F,0,+). — hadadrefael, Oct 22 '23 at 10:07
@Asigan A group has one 0-ary operation, one 1-ary operation, and one 2-ary operation. Not just a 2-ary operation. — Martin Brandenburg, Oct 22 '23 at 10:13

Martin Brandenburg · Accepted Answer · 2023-10-22T10:20:28.873

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You ask if the additive group of a field only trivial normal subgroups. But every subgroup is normal (the group is commutative), so you ask if the additive group has only trivial subgroups.

Well, a group has only the trivial subgroups iff it is trivial or isomorphic to $\mathbb{Z}/p\mathbb{Z}$ for a prime number $p$ (SE/1516797).

So the answer to your question is "No". Take any infinite field, for example.

But there is a similarity between ideals and normal subgroups. Namely, ideals of a ring $R$ classify the surjective ring homomorphisms $R \to S$ (take the kernel of a ring homomorphism). And normal subgroups of a group $G$ classify the surjective group homomorphisms $G \to H$ (take the kernel of a group homomorphism).

More generally, if $\mathcal{T}$ is any algebraic theory (this won't make much sense to you right now, but maybe later), then the congruence relations on a $\mathcal{T}$-algebra $A$ classify the surjective $\mathcal{T}$-algebra homomorphisms $A \to B$. The congruence relations on a ring correspond to the ideals, the congruence relations on a group correspond to the normal subgroups. Therefore, from this perspective of universal algebra, "normal subgroups" and "ideals" are the same concept.

edited Oct 22 '23 at 10:20

answered Oct 22 '23 at 10:15

Martin Brandenburg

163,620

can we say something about finite fields, then ? – hadadrefael Oct 22 '23 at 10:17
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No. Consider $\mathbb{F}_{p^2}$. – Martin Brandenburg Oct 22 '23 at 10:17
understood. thank you! – hadadrefael Oct 22 '23 at 10:20
Congruence relations correspond to ideals only for very special types of algebras - see here. – Bill Dubuque Oct 22 '23 at 17:00
I only talked about rings when talking about ideals. – Martin Brandenburg Oct 22 '23 at 17:14

Similarities Between Rings and Groups: Ideals and Normal Subgroups

1 Answers1