regular ideal

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

Proposition.  If m is a positive integer, then the only regular ideal in the residue class ring m is the unit ideal (1).

Proof.  The ring m is a principal ideal ring.  Let (n) be any regular ideal of the ring m.  Then n can not be zero divisor, since otherwise there would be a non-zero element r of m such that  nr=0  and thus every element sn of the principal idealMathworldPlanetmathPlanetmath would satisfy  (sn)r=s(nr)=s0=0.  So, n is a regular element of 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 congruenceMathworldPlanetmath1yn(modm),  i.e.  1=yn  in the ring m.  With  1 the principal ideal (n) contains all elements of m, which means that  (n)=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 ringMathworldPlanetmath R/I is a regular ringMathworldPlanetmath, in other words, if for each  aR  there exists an element  bR  such that  a2b-aI.


  • 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
Canonical name RegularIdeal
Date of creation 2013-03-22 15:43:05
Last modified on 2013-03-22 15:43:05
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Definition
Classification msc 14K99
Classification msc 16D25
Classification msc 11N80
Classification msc 13A15
Related topic QuasiRegularIdeal
Related topic QuasiRegularity