Ring
Definition of a Ring:

Set and Operations: A ring $(R,+,\cdot )$ consists of a set $R$ equipped with two binary operations, addition $+$ and multiplication $\cdot $, satisfying the following properties:

Additive Group: $(R,+)$ is an abelian group.
 Closure under addition: For all $a,b$ in $R$, $a+b$ is in $R$.
 Associativity: For all $a,b,c$ in $R$, $(a+b)+c=a+(b+c)$.
 Identity element (zero): There exists an element $0$ in $R$ such that for any $a$ in $R$, $a+0=0+a=a$.
 Inverse element: For each $a$ in $R$, there exists an element $a$ in $R$ such that $a+(a)=(a)+a=0$.
 Commutativity: For all $a,b$ in $R$, $a+b=b+a$.

Multiplication: The multiplication operation is associative.
 Distributive laws hold: For all $a,b,c$ in $R$, $a\cdot (b+c)=a\cdot b+a\cdot c$ and $(a+b)\cdot c=a\cdot c+b\cdot c$.

Key Concepts:

Zero and Unity: The additive identity element is denoted as $0$, and if a multiplicative identity element exists in the ring, it's called unity, often denoted as $1$.

Commutative Ring: If the multiplication operation in a ring is commutative (i.e., $a\cdot b=b\cdot a$ for all $a,b$ in $R$), the ring is called commutative.

Ring with Unity: A ring $R$ is said to have a unity if there exists an element $1$ in $R$such that for any $a$ in $R$, $a\cdot 1=1\cdot a=a$.

Zero Divisors: Elements $a,b$ in a ring $R$ are zero divisors if $a\mathrm{\ne}0$, $b\mathrm{\ne}0$, and $a\cdot b=0$. Rings without zero divisors are called integral domains.
Examples:

Integers: The set of integers $\mathbb{Z}$ forms a ring under addition and multiplication. It is both commutative and has unity.

Polynomial Rings: The set of polynomials with coefficients from a ring forms another ring under polynomial addition and multiplication.

Matrix Rings: The set of $n\times n$ matrices with entries from a ring $R$ forms a ring under matrix addition and multiplication.
Ring Properties:

Associative Property: Both addition and multiplication in a ring are associative.

Distributive Property: Multiplication distributes over addition, i.e., $a\cdot (b+c)=a\cdot b+a\cdot c$ and $(a+b)\cdot c=a\cdot c+b\cdot c$.

Ring Homomorphism: A function between two rings $f:R\to S$ is a ring homomorphism if it preserves addition and multiplication.

Subrings: A subset $S$ of a ring $R$ is a subring if $S$ is itself a ring under the same operations as $R$.

Characteristic of a Ring: The smallest positive integer $n$ such that $n\cdot 1=0$ in a ring $R$ is called the characteristic of $R$. If no such $n$ exists, the characteristic is considered to be zero.