Definition 5.1. A ring a triple $(R,+,\cdot)$ consisting of a set $R$, an addition $+$, and a multiplication $\cdot$, such that the following axioms are verified:

(1) $(R,+)$ is an abelian additive group;

(2) Associativity: $(xy)z=x(yz)$ for all $x,y,z\in R$;

(3) Distributivity: $x(y+z)=xy+xz$ and $(x+y)z=xz+yz$ for all $x,y,z\in R$;

We will simply denote by $R$ when there is no confusion. We say that $R$ is commutative if $xy=yx$ for all $x,y\in R$. And we say that $R$ is a ring with identity if there exists a multiplicative identity in $(R,\cdot)$, which is denoted by $1$. And we denote the additive identity in $(R,+)$ by $0$ and call it the zero element of $R$. The ring consisting of only 0 is called the null ring, which is also denoted by $0$.

Example 5.2. $(\mathbb{Z},+,\cdot)$, $(\mathbb{Q},+,\cdot)$, $(\mathbb{R},+,\cdot)$, $(\mathbb{C},+,\cdot)$, $(\mathbb{Z}/n\mathbb{Z},+,\cdot)$ are all rings with identity.

Definition 5.3. A homomorphism of rings $f:R\rightarrow S$ is a function $R\rightarrow S$ such that $f(x+y)=f(x)+f(y)$ and $f(xy)=f(x)f(y)$ for all $x,y\in R$. Moreover, if $R,S$ are rings with identity, then the homomorphism $f$ also preserves identity: $f(1)=1$. A homomorphism of rings is an isomorphism if it is bijective.

Definition 5.4. Let $R$ be a ring. A subring of $R$ is a subset $S\subset R$ such that $S$ forms a ring under the induced operations from $R$. An ideal of $R$ is a subset $I\subset R$ such that $(I,+)$ is an additive subgroup and $xy\in I,yx\in I$ for all $x\in R,y\in I$.

Definition 5.5. An ideal $I$ of a ring $R$ is a maximal ideal if for an ideal $J\subset R$, $I\subset J$ implies $I=J$.

Definition/Proposition 5.6. Let $R$ be a ring and let $I$ be an ideal of $R$. The set of all cosets of $(I,+)$ forms a ring under the operations $(x+I)+(y+I)=(x+y)+I$ and $(x+I)(y+I)=xy+I$ for all $x,y\in R$. Such a ring is called the quotient ring of $R$ by $I$, and is denoted by $R/I$.

Example 5.7. The ring of integers modulo $n$ for some $n\in\mathbb{N}$, denoted by $\mathbb{Z}_{n}$, is the quotient ring $\mathbb{Z}/n\mathbb{Z}$ of $\mathbb{Z}$.

Definition 5.8. Let $R$ be a ring with identity. If there exists a least positive integer $n$ such that $n1=0$, then we say that $R$ has characteristic $n$. If there for all integers $n>0$, $n1\neq0$, then we say that $R$ has characteristic 0.

Definition 5.9. An integral domain or simply a domain is a commutative ring $R$ with identity such that $xy\neq0$ for all $x\neq0,y\neq0$ in $R$.

Definition 5.10. An ideal $\mathfrak{p}$ of a ring $R$ is a prime ideal if $xy\in \mathfrak{p}$ implies $x\in\mathfrak{p}$ or $y\in\mathfrak{p}$ for all $x,y\in R$.

Proposition 5.11. Let $R$ be a commutative ring with identity and let $I$ be an ideal of $R$. Then $R/I$ is a domain if and only if $I$ is prime. And $R/I$ is a field if and only if $I$ is a maximal ideal. Consequently, every maximal ideal of $R$ is prime.

Definition 5.12. An ideal is principle if it is generated by a single element.

Definition 5.13. A principle ideal domain or PID for short is a domain whose ideals are all principle ideals.

Example 5.14. The ring $\mathbb{Z}$ and the polynomial ring $K[X]$ for a field $K$ are PIDs.