An ideal of a ring is called a , iff a regular element of .
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 (http://planetmath.org/BezoutsLemma), 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. . There is also a different notion of “regular ideal” mentioned in  (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 .
- 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).
|Date of creation||2013-03-22 15:43:05|
|Last modified on||2013-03-22 15:43:05|
|Last modified by||pahio (2872)|