# Ring

Definition of a Ring:

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

1. Additive Group: $\left(R,+\right)$ 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 $R$, $\left(a+b\right)+c=a+\left(b+c\right)$.
• 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 $a+\left(-a\right)=\left(-a\right)+a=0$.
• Commutativity: For all $a,b$ in $R$, $a+b=b+a$.
2. Multiplication: The multiplication operation is associative.

• Distributive laws hold: For all $a,b,c$ in $R$, $a\cdot \left(b+c\right)=a\cdot b+a\cdot c$ and $\left(a+b\right)\cdot c=a\cdot c+b\cdot c$.

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 $n×n$ 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., $a\cdot \left(b+c\right)=a\cdot b+a\cdot c$ and $\left(a+b\right)\cdot c=a\cdot c+b\cdot c$.

• 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.