In mathematics, more specifically abstract algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exists a and b in R with a·b ≠ b·a.

What is noncommutative?

Informally, noncommutative means "order matters". More formally, if is some binary operation on a set, and and are elements of that set, then noncommutative means that doesn't necessarily equal . Most common operations, such as addition and multiplication of numbers, are commutative.

Is Z is non commutative ring?

The integers Z with the usual addition and multiplication is a commutative ring with identity. The only elements with (multiplicative) inverses are ±1. The integers modulo n: Z_n form a commutative ring with identity under addition and multiplication modulo n.