Integral Domain

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April 15th 2009, 06:01 PM
chiph588@

Integral Domain

If $R$ is an integral domain and $a, b \in R$ , show if $Ra = Rb$ then $a=ub$ where $u$ is a unit.
April 15th 2009, 06:20 PM
ThePerfectHacker

Quote:

Originally Posted by chiph588@

If $R$ is an integral domain and $a, b \in R$ , show if $Ra = Rb$ then $a=ub$ where $u$ is a unit.

Since $(a) = (b)$ it means $(a)\subseteq (b) \text{ and }(b) \subseteq (a)$ .
Thus, $b|a \text{ and } a|b$ so $a,b$ are associates and so $a=ub$ .
April 15th 2009, 07:33 PM
NonCommAlg

Quote:

Originally Posted by chiph588@

If $R$ is an integral domain and $a, b \in R$ , show if $Ra = Rb$ then $a=ub$ where $u$ is a unit.

to see why we need R to be an integral domain:

if $a=0,$ then $Rb=0$ and so $b=0.$ in this case choose $u=1.$ so we may assume that $a \neq 0.$ now we have $a \in Ra = Rb.$ so $a=rb,$ for some $r \in R.$ similarly $b=sa,$ for some $s \in R.$

thus $a=rb=rsa.$ hence $(1-rs)a=0.$ since $R$ is an integral domain and $a \neq 0,$ we must have $1-rs=0,$ i.e. $rs=1.$ so $r$ is a unit and we choose $u=r.$

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