# nested ideals in von Neumann regular ring

###### Theorem.

Let $\mathfrak{a}$ be an ideal of the von Neumann regular ring $R$. βThen $\mathfrak{a}$ itself is a von Neumann regular ring and any ideal $\mathfrak{b}$ of $\mathfrak{a}$ is likewise an ideal of $R$.

###### Proof.

If β$a\in\mathfrak{a}$, then β$asa=a$β for some β$s\in R$. βSetting β$t=sas$β we see that $t$ belongs to the ideal $\mathfrak{a}$ and

 $ata=a(sas)a=(asa)sa=asa=a.$

Secondly, we have to show that whenever β$b\in\mathfrak{b}\subseteq\mathfrak{a}$β and β$r\in R$, then both $br$ and $rb$ lie in $\mathfrak{b}$. βNow, β$br\in\mathfrak{a}$β because $\mathfrak{a}$ is an ideal of $R$. βThus there is an element $x$ in $\mathfrak{a}$ satisfying β$brxbr=br$. βSince $rxbr$ belongs to $\mathfrak{a}$ and $\mathfrak{b}$ is assumed to be an ideal of $\mathfrak{a}$, we conclude that the product β$b\cdot rxbr$β must lie in $\mathfrak{b}$, i.e. β$br\in\mathfrak{b}$. βSimilarly it can be shown that β$rb\in\mathfrak{b}$. β

## References

• 1 David M. Burton: A first course in rings and ideals. βAddison-Wesley. βReading, Massachusetts (1970).
Title nested ideals in von Neumann regular ring NestedIdealsInVonNeumannRegularRing 2013-03-22 14:48:24 2013-03-22 14:48:24 CWoo (3771) CWoo (3771) 12 CWoo (3771) Theorem msc 16E50