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

Proposition. If is a positive integer, then the only regular ideal in the residue class ring is the unit ideal .

Proof. The ring is a principal ideal ring. Let be any regular ideal of the ring . Then can not be zero divisor, since otherwise there would be a non-zero element of such that and thus every element of the principal ideal would satisfy . So, is a regular element of and therefore we have . Then, according to Bézout's lemma, there are such integers and that . This equation gives the congruence , i.e. in the ring . With the principal ideal contains all elements of , which means that .

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 be an ideal of the commutative ring with non-zero unity. This ideal is called regular, if the quotient ring is a regular ring, in other words, if for each there exists an element such that .

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).