----- $\mathsf{Ab}$ denotes the [[category]] of [[abelian group|abelian groups]]. > [!proposition] Proposition. ([[endsets in ab are rings]]) > For every [[abelian group]] $G$, the [[group]] $\text{End}_{\mathsf{Ab}}(G):=\text{Hom}_{\mathsf{Ab}}(G,G)$ forms a [[ring]] under the operations of (function) addition ([[homsets in ab are abelian groups|as here]]) and composition, called the **endomorphism ring** of $G$. > > Especially important is the [[endomorphism ring of the group of integers under addition is isomorphic to the ring of integers|endomorphism ring]] of the [[integers under addition|integers]] $(\mathbb{Z}, +)$. ^proposition > [!basicproperties] > - The [[unit|group of units]] in $\text{End}_{\mathsf{Ab}}(G)$ is, by definition, the subset of elements having two-sided [[inverse map|inverse]] — i.e., the [[automorphism|automorphism group]] $\text{Aut}_{\mathsf{Ab}}(G)$. ^properties > [!proof]- Proof. ([[endsets in ab are rings]]) > The underlying [[abelian group]] structure is derived in [[homsets in ab are abelian groups]], so we just have to check that the added multiplication [[binary operation|operation]] behaves well. Associativity is immediate since from the [[category]] axioms, and distributivity of multiplication over addition is immediate. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```