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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 divisor2.

A commutative ring \(R\), which is not the zero ring and in which \(0\) is the only zero divisor3 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


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Bibliography (further reading)

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

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