---- > [!definition] Definition. ([[ring]]) > A **ring** $(R,+,\cdot)$ is an [[abelian group]] $(R,+)$ endowed with a second [[binary operation]] $\cdot$, satisfying on its own the requirements of being [[associative|associative]] and have a two-sided identity, i.e., > - For all $r,s,r \in R$ we have $(r \cdot s) \cdot t=r \cdot (s \cdot t)$; > - There exists $1_{R} \in R$ such that for all $r \in R$, $r \cdot 1_{R}=r=1_{R} \cdot R$ > (which make $(R,\cdot)$ a [[monoid]]), and further interacting with $+$ via the following [[distributive property|distributive properties]]: for all $r,s,t, \in R$, $(r+s) \cdot t = r \cdot t + s \cdot t \text{ and } t \cdot (r + s)=t \cdot r + t \cdot s.$ > > > A **rng** is a ring without identity. > > Rings are objects of [[category]] $\mathsf{Ring}$. The morphism set $\text{Hom}(R,S)$ is the set of [[ring homomorphism|ring homomorphisms]] from $R$ to $S$. ^definition > [!basicproperties] The category $\mathsf{Ring}$. > 1. The zero-ring is (clearly) [[terminal object|final]] in $\mathsf{Ring}$. It is not [[terminal object|initial]], however: because [[ring homomorphism|ring homomorphisms]] preserve both additive/multiplicative identity, the only rings to which zero rings map homomorphically are the zero-rings. ^835ab0 >2. Instead, [[terminal object|initial]] in $\mathsf{Ring}$ are the integers $(\mathbb{Z}, +, \cdot)$, via the map $n \xmapsto{\varphi}n \cdot 1_{R} = \overbrace{1_{R} + \dots + 1_{R}}^{n \text{ times}}$. $\varphi$ is a [[group homomorphism]] because $\varphi(n+m)=(n+m) \cdot 1_{R}=n\cdot 1_{R} + m \cdot 1_{R}$; it is also a [[ring homomorphism]] because $\varphi(nm)=(nm) \cdot 1_{R}=( n 1_{R}) \cdot (m 1_{R})$ and (trivially) $1_{R}$ is preserved. This [[ring homomorphism]] is unique, since it is determined by the identity-preservation and addition-preservation requirement. ^properties ---- #### ---- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag