An ideal $\mathfrak{a}$ of a ring $R$ is called a regular, iff $\mathfrak{a}$ contains 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, 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 $a^2b\!-\!a \in I$ .

Bibliography

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).