# Ring

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Definition
 Ring A non empty set $R$ is said to be a Ring with respect to two binary operations $+$ (called as addition) and $.$ (called as multiplication) if it satisfies the following conditions: $(R, +)$ is an abelian group, multiplication is associative: i.e. $x(yz) = (xy)z$ for all $x, y, z \in R$, the distributive laws: $(x + y)z = x.z + y.z$ and $x(y + z) = x.y + x.z$ holds true for all $x, y, z \in R$ Ring with identity: If multiplication identity 1 exists then we say that $R$ is a ring with unity. Commutative Ring: If multiplication in $R$ is commutative then we say that $R$ is a commutative ring. Unit Element: An element a in $R$ is said to be a unit if there exists $b \in R$ such that $ab = ba = 1$. The set of all units in $R$ is denoted by $U(R)$. The set $U(R)$ forms a group under multiplication. Division Ring: If every non-zero element of $R$ is a unit then $R$ is called as a division ring. Field: A commutative division ring is called as a Field.

## Examples

1. The set of integers $\mathbb Z$, the set of rationals $\mathbb Q$, the set of reals $\mathbb R$, the set of complex numbers $\mathbb C$ are all examples of commutative rings with identity. The sets $\mathbb Q, \mathbb R$ are division rings.
2. Matrix Rings: The set Mn(R) which denotes the set of all $n\times n$ matrices with entries from ring R is called as a matrix ring with respect to usual addition and multiplication.