**Definition**: Unit and Unit Group, Zero Divisor and Integral Domain

Let \((R, + ,\cdot)\) be a ring with \(1\) as the identity of \((R,\cdot)\) and \(0\) as the identity of \((R,+)\). Further, let \(a\in R\).

- If there exists an \(b\in R\) with \(a\cdot b=1\), then we call \(a\) a
**unit**in \(R\) (i.e. the units of a ring are all the elements which have a multiplicative inverse). - The group \((R,\cdot)\) is called the
**unit group**of \(R\) and denoted by \(R^*\).^{1} - If \(a\neq 0\) and there exists \(b\in R, b\neq 0\) with
- \(a\cdot b=0\), then we call \(a\) a
**left zero divisor**in \(R\). - \(b\cdot a=0\), then we call \(a\) a
**right zero divisor**in \(R\). - we call \(a\) a
**zero divisor**in \(R\), if \(a\) is both, a left and a right zero divisor^{2}.

- \(a\cdot b=0\), then we call \(a\) a

A commutative ring \(R\), which is not the zero ring and in which \(0\) is the only zero divisor^{3} is called **integral domain**.

^{1} Please note that unities in \(R\) are exactly those of its elements, which have inverse elements with respect to the operation “\(\cdot\)”. Since, by definition of a ring and in general, \((R,\cdot)\) is “only” a monoid (and not a group), unity elements are those elements in the monoid \((R,\cdot)\), which make it “behave” like a group.

^{2} Please note that if \(R\) is commutative, every left zero divisor is also a right zero divisor.

^{3} Please note that this is equivalent to \(a\cdot b=0\Leftrightarrow a=0 \vee b=0\).

| | | | | created: 2014-08-31 13:20:40 | modified: 2016-10-10 14:15:12 | by: *bookofproofs* | references: [677]

## 1.**Proposition**: Cancellation Law

## 2.**Definition**: Zero Ring

[677] **Modler, Florian; Kreh, Martin**: “Tutorium Algebra”, Springer Spektrum, 2013

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