# regular ideal

An ideal $\mathfrak{a}$ of a ring $R$ is called a , iff $\mathfrak{a}$ a regular element of $R$.

If $m$ is a positive integer, then the only regular ideal in the residue class ring $\mathbb{Z}_{m}$ is the unit ideal $(1)$.

Proof.  The ring $\mathbb{Z}_{m}$ is a principal ideal ring.  Let $(n)$ be any regular ideal of the ring $\mathbb{Z}_{m}$.  Then $n$ can not be zero divisor, since otherwise there would be a non-zero element $r$ of $\mathbb{Z}_{m}$ such that  $nr=0$  and thus every element $sn$ of the principal ideal would satisfy  $(sn)r=s(nr)=s0=0$.  So, $n$ is a regular element of $\mathbb{Z}_{m}$ and therefore we have  $\gcd(m,\,n)=1$.  Then, according to Bézout’s lemma (http://planetmath.org/BezoutsLemma), there are such integers $x$ and $y$ that  $1=xm\!+\!yn$.  This equation gives the congruence$1\equiv yn\pmod{m}$,  i.e.  $1=yn$  in the ring $\mathbb{Z}_{m}$.  With  $1$ the principal ideal $(n)$ contains all elements of $\mathbb{Z}_{m}$, which means that  $(n)=\mathbb{Z}_{m}=(1)$.

Note.  The above notion of “regular ideal” is used in most books concerning ideals of commutative rings, e.g. [1].  There is also a different notion of “regular ideal” mentioned in [2] (p. 179):  Let $I$ be an ideal of the commutative ring $R$ with non-zero unity.  This ideal is called regular, if the quotient ring $R/I$ is a regular ring, in other words, if for each  $a\in R$  there exists an element  $b\in R$  such that  $a^{2}b\!-\!a\in I$.

## References

• 1 M. Larsen and P. McCarthy:Multiplicative theory of ideals”.  Academic Press. New York (1971).
• 2 D. M. Burton:A first course in rings and ideals”.  Addison-Wesley. Reading, Massachusetts (1970).
Title regular ideal RegularIdeal 2013-03-22 15:43:05 2013-03-22 15:43:05 pahio (2872) pahio (2872) 12 pahio (2872) Definition msc 14K99 msc 16D25 msc 11N80 msc 13A15 QuasiRegularIdeal QuasiRegularity