Definition of Unit in the Ring

Question

$ Definition $

$ Unity$

A $Unity$ in a ring is a Nonzero element that is an identity under multiplication.

$Unit$

A Nonzero element of a $ commutative$ ring with a multiplicative inverse is called $Unit$ of a ring.

$Doubt$

Is it necessary to have a commutative ring to define Unit of a ring ?

Answer 1

You do not need commutativity to define a unit. However, the multiplicative inverse of an element must necessarily commute with that element. That is, if $ u \in R $ is a unit and $ v $ is its inverse, that by definition means $ uv = 1 = vu $. This is the same as the defintion of the inverse in a (not necessarily abelian) group.

Answer 2

You don't need commutativity for defining a unit. You need to be sure, however, to define the multiplicative inverse of $u$ as an element $v$ such that $$ uv=1=vu $$ Just requiring one of these is not sufficient. You can try your hand into finding a ring where there is an element $u$ with $uv=1$ for some $v$, but $xu\ne1$ for every $x$ (so $u$ is right invertible but not left invertible).

Obviously, if the ring is commutative requiring $uv=1$ suffices.