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A non empty set [math]R[/math] is said to be a Ring with respect to two binary operations [math]+[/math] (called as addition) and [math].[/math] (called as multiplication) if it satisfies the following conditions:
- [math](R, +)[/math] is an abelian group,
- multiplication is associative: i.e. [math] x(yz) = (xy)z[/math] for all [math]x, y, z \in R [/math],
- the distributive laws: [math](x + y)z = x.z + y.z[/math] and [math] x(y + z) = x.y + x.z [/math] holds true for all [math] x, y, z \in R[/math]
Ring with identity:
If multiplication identity 1 exists then we say that [math]R[/math] is a ring with unity.
If multiplication in [math]R[/math] is commutative then we say that [math]R[/math] is a commutative ring.
An element a in [math]R[/math] is said to be a unit if there exists [math] b \in R [/math] such that [math] ab = ba = 1 [/math].
The set of all units in [math]R[/math] is denoted by [math]U(R)[/math]. The set [math]U(R)[/math] forms a group under multiplication.
Division Ring: If every non-zero element of [math]R[/math] is a unit then [math]R[/math] is called as a division ring.
Field: A commutative division ring is called as a Field.
- The set of integers [math] \mathbb Z [/math], the set of rationals [math] \mathbb Q [/math], the set of reals [math] \mathbb R [/math], the set of complex numbers [math]\mathbb C[/math] are all examples of commutative rings with identity. The sets [math]\mathbb Q, \mathbb R[/math] are division rings.
- Matrix Rings: The set Mn(R) which denotes the set of all [math]n\times n [/math] matrices with entries from ring R is called as a matrix ring with respect to usual addition and multiplication.