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Revision difference : regular ideal |
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An ideal $\mathfrak{a}$ of a ring is called a {\em regular}, iff $\mathfrak{a}$ \PMlinkescapetext{contains} a regular element.
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An ideal $\mathfrak{a}$ of a ring is called a {\em regular ideal}, iff $\mathfrak{a}$ \PMlinkescapetext{contains} a regular element.
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| \textbf{Proposition.}\, If $m$ is a positive integer, then the only regular ideal in the residue class ring $\mathbb{Z}_m$ is the unit ideal $(1)$. |
\textbf{Proposition.}\, If $m$ is a positive integer, then the only regular ideal in the residue class ring $\mathbb{Z}_m$ is the unit ideal $(1)$. |
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| {\em 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 \PMlinkname{B\'ezout's lemma}{BezoutsLemma}, 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\, $(n) = \mathbb{Z}_m = (1)$. |
{\em 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 \PMlinkname{B\'ezout's lemma}{BezoutsLemma}, 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\, $(n) = \mathbb{Z}_m = (1)$. |
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| \textbf{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 aregular ring, in other words, if for each\, $a \in R$\, there exists an element\, $b \in R$\, such that\, |
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| $a^2b\!-\!a \in I$. |
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| \begin{thebibliography}{9} |
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| \bibitem{LM}{\sc M. Larsen and P. McCarthy:} ``{\em Multiplicative theory of ideals}''.\, Academic Press. New York (1971). |
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| \bibitem{B}{\sc D. M. Burton:} ``{\em A first course in rings and ideals}''.\, Addison-Wesley. Reading, Massachusetts (1970). |
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| \end{thebibliography} |
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