Let’s keep stock of what we’ve covered so far for ring theory, and compare it to the case of groups. There are loads of parallels between the two cases.

G is a group |
R is a ring. |

Abelian groups. | Commutative rings. |

Group products. | Ring products. |

is a subgroup. | is a subring. |

Intersection of subgroups is a subgroup. | Intersection of subrings is a subring. |

Next, we look at normal subgroups of a group and ideals of a ring.

Normal subgroups. | Ideals. |

If N is normal in G, then G/N is a group quotient. |
If I is an ideal of R, then R/I is a ring quotient. |

Intersection of normal subgroups is a normal subgroup. | Intersection of ideals is an ideal. |

If is a subgroup and N is a normal subgroup of G, then is a normal subgroup of H. |
If is a subring and I is an ideal of R, then is an ideal of S. |

If is a subgroup and is a normal subgroup, then HN is a subgroup of G |
If is a subring and is an ideal, then I+S is a subring of R. |

If are normal subgroups, then so is . | If are ideals, then so is . |

?? | If are ideals, then so is . |

Finally, there’re group and ring homomorphisms.

f : G → H is a group homomorphism. |
f : R → S is a ring homomorphism. |

Composing group homomorphisms gives a group homomorphism. | Composing ring homomorphisms gives a ring homomorphism. |

If is a subgroup, then so is . | If is a subring, then so is . |

If is a subgroup, then so is . | If is a subring, then so is . |

If is a normal subgroup, then so is . | If is an ideal, then so is . |

First isomorphism theorem: as groups. | First isomorphism theorem: as rings. |

Second isomorphism theorem: let be a subgroup and be a normal subgroup. Then . | Second isomorphism theorem: let be a subring and be an ideal. Then . |

Third isomorphism theorem: let , where N and H are normal subgroups of G. Then . |
Third isomorphism theorem: let , where I and J are ideals of R. Then . |