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## Definition: Unit

Let $(R,\cdot,+)$ be an integral domain with the multiplicative neutral element $1,$ and let $a\in R.$

We call $a$ a unit of $R$ if and only if $$a\mid 1\,$$ i.e. $a$ is a divisor of $1$.

### Notes

• Unfolding the definition of divisor in a ring, this means that there exists $$b\in R$$ with $$a\cdot b=1$$.
• In other words, units in $$R$$ are exactly those of its elements, which have inverse elements with respect to the operation “$$\cdot$$”.

| | | | | created: 2019-06-27 18:36:41 | modified: 2019-06-27 18:48:22 | by: bookofproofs | references: [8250]

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