## Introduction to Ring Theory (6)

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. $H\subseteq G$ is a subgroup. $S\subseteq R$ 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 $H\subseteq G$ is a subgroup and N is a normal subgroup of G, then $H\cap N$ is a normal subgroup of H. If $S\subseteq R$ is a subring and I is an ideal of R, then $S\cap I$ is an ideal of S. If $H\subseteq G$ is a subgroup and $N\subseteq G$ is a normal subgroup, then HN is a subgroup of G If $S\subseteq R$ is a subring and $I\subseteq R$ is an ideal, then I+S is a subring of R. If $N_1, N_2\subseteq G$ are normal subgroups, then so is $N_1 N_2$. If $I_1, I_2\subseteq R$ are ideals, then so is $I_1 + I_2$. ?? If $I_1, I_2\subseteq R$ are ideals, then so is $I_1 I_2$.

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 $K\subseteq G$ is a subgroup, then so is $f(K)\subseteq H$. If $T\subseteq R$ is a subring, then so is $f(T)\subseteq S$. If $L\subseteq H$ is a subgroup, then so is $f^{-1}(L)\subseteq G$. If $U\subseteq S$ is a subring, then so is $f^{-1}(U)\subseteq R$. If $N\subseteq H$ is a normal subgroup, then so is $f^{-1}(N)\subseteq G$. If $J\subseteq S$ is an ideal, then so is $f^{-1}(J)\subseteq R$. First isomorphism theorem: $G/\text{ker}(f)\cong \text{im}(f)$ as groups. First isomorphism theorem: $R/\text{ker}(f)\cong \text{im}(f)$ as rings. Second isomorphism theorem: let $H\subseteq G$ be a subgroup and $N\subseteq G$ be a normal subgroup. Then $H/(H\cap N)\cong HN/N$. Second isomorphism theorem: let $S\subseteq R$ be a subring and $I\subseteq R$ be an ideal. Then $S/(S\cap I) \cong (S+I)/I$. Third isomorphism theorem: let $N\subseteq H\subseteq G$, where N and H are normal subgroups of G. Then $(G/N)/(H/N) \cong G/H$. Third isomorphism theorem: let $I\subseteq J\subseteq R$, where I and J are ideals of R. Then $(R/I)/(J/I) \cong R/J$.
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