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[parent] regular ideal (Definition)

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).



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See Also: quasi-regular ideal, quasi-regularity

Also defines:  regular ideal

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Cross-references: regular ring, quotient ring, regular, non-zero unity, commutative rings, contains, congruence, equation, principal ideal, zero divisor, principal ideal ring, unit ideal, residue class ring, integer, positive, proposition, regular element, iff, ring, ideal
There are 6 references to this entry.

This is version 8 of regular ideal, born on 2006-03-03, modified 2006-06-19.
Object id is 7666, canonical name is RegularIdeal.
Accessed 1430 times total.

Classification:
AMS MSC13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)
 11N80 (Number theory :: Multiplicative number theory :: Generalized primes and integers)
 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
 14K99 (Algebraic geometry :: Abelian varieties and schemes :: Miscellaneous)

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