Welcome guest
You're not logged in.
190 users online, thereof 0 logged in

## 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]

(none)