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$ 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 ?

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    $\begingroup$ In a word...no. $\endgroup$ Jul 9, 2018 at 5:11
  • $\begingroup$ Ok thanks...... $\endgroup$
    – blue boy
    Jul 9, 2018 at 5:23

2 Answers 2

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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.

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  • $\begingroup$ Ok. Thanks. I got confused. This is the way Unit is defined in the Gallian. $\endgroup$
    – blue boy
    Jul 9, 2018 at 5:23
  • $\begingroup$ So Unit is any element that has inverse $\endgroup$
    – blue boy
    Jul 9, 2018 at 5:30
  • $\begingroup$ Correct, and an element commutes with its inverse, but you don't need commutativity of the ring to define this. $\endgroup$ Jul 9, 2018 at 5:31
  • $\begingroup$ Ok thanks...... $\endgroup$
    – blue boy
    Jul 9, 2018 at 5:32
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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.

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