regular ideal

An ideal $𝔞$ of a ring $R$ is called a , iff $𝔞$ a regular element of $R$.

Proposition.  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  $1\equiv yn(modm)$,  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 $a^{2}b-a\in I$.