## You are here

Homeregular ideal

## Primary tabs

# regular ideal

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 $1\equiv yn\;\;(\mathop{{\rm mod}}m)$, 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).

## Mathematics Subject Classification

14K99*no label found*16D25

*no label found*11N80

*no label found*13A15

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections