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I was studying my abstract algebra notes, and couldn't help but notice a striking similarity between the following two statements:

  1. Let $G$ be a group, and $H\triangleleft G$. The canonical projection $$\pi:G\to G/H$$ induces a bijection between subgroups of $G$ that contain $H$, and subgroups of $G/H$. As a corollary, $G/H$ is simple iff $H$ is maximal among proper normal subgroups of $G$.

  2. Let $R$ be a ring, and $\mathfrak m$ an ideal. The canonical projection $$\pi:R\to R/\mathfrak m$$ induces a bijection between ideals in $R$ that contain $\mathfrak m$, and ideals of $R/\mathfrak m$. As a corollary, $R/\mathfrak m$ is a field iff $m$ is a maximal ideal.

There has to be a relationship between the two, right? I might even go so far as to say that $1. \Leftrightarrow 2.$ (and obviously $\text{true}\Leftrightarrow\text{true}$ always holds but, well, you know what I mean). Can someone shed some light?

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Here are a few ways that those facts are connected.

  1. Rings are groups with extra structure. This fact is how you prove isomorphism theorems for rings, for the most part: use the corresponding group theorem for the additive group of the ring, and then check that the other properties that need to be true are true.

  2. This lies under the umbrella of universal algebra: there is a way to generalize the idea of a group or a ring to other structures, and in these structures, generalized versions of the isomorphism theorems hold. (See the Wikipedia page on this subject.) You'll see another instance of the theorems when you learn about modules.

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