| Version 18 |
Version 17 |
| If $a$ and $b$ are two positive integers, then their {\em least common multiple}, denoted by $\lcm(a, b)$, is the positive integer $f$ satisfying the conditions |
If $a$ and $b$ are two positive integers, then their {\em least common multiple}, denoted by $\lcm(a, b)$, is the positive integer $f$ satisfying the conditions |
| \begin{itemize} |
\begin{itemize} |
| \item $a\mid f$ and $b\mid f$, |
\item $a\mid f$ and $b\mid f$, |
| \item if $a\mid c$ and $b\mid c$, then $f\mid c$. |
\item if $a\mid c$ and $b\mid c$, then $f\mid c$. |
| \end{itemize} |
\end{itemize} |
| Note: The definition can be generalized for several numbers. The positive $\lcm$ of positive integers is uniquely determined. (Its negative satisfies the same two conditions.) |
Note: The definition can be generalized for several numbers. The positive $\lcm$ of positive integers is uniquely determined. (Its negative satisfies the same two conditions.) |
| \subsection*{Properties} |
\subsection*{Properties} |
| \begin{enumerate} |
\begin{enumerate} |
| \item If $a = \prod_{i=1}^{m}p_i^{\alpha_i}$ and |
\item If $a = \prod_{i=1}^{m}p_i^{\alpha_i}$ and |
| $b = \prod_{i=1}^{m}p_i^{\beta_i}$ are the prime factor presentations of the positive integers $a$ and $b$ ($\alpha_{i} \ge 0$, $\beta_{i} \ge 0$ $\forall i$), then |
$b = \prod_{i=1}^{m}p_i^{\beta_i}$ are the prime factor presentations of the positive integers $a$ and $b$ ($\alpha_{i} \ge 0$, $\beta_{i} \ge 0$ $\forall i$), then |
| $$\lcm(a, b)= \prod_{i=1}^{m}p_i^{\max(\alpha_i, \beta_i)}.$$ |
$$\lcm(a, b)= \prod_{i=1}^{m}p_i^{\max(\alpha_i, \beta_i)}.$$ |
| This can be generalized for $\lcm$ of several numbers. |
This can be generalized for $\lcm$ of several numbers. |
| \item $$\gcd(a, b)\cdot \lcm(a, b) = ab$$ |
\item $$\gcd(a, b)\cdot \lcm(a, b) = ab$$ |
| This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example, |
This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example, |
| $$\gcd(a, b, c)\cdot \lcm(a, b, c) \neq abc.$$ |
$$\gcd(a, b, c)\cdot \lcm(a, b, c) \neq abc.$$ |
| \item The preceding formula may be presented in terms of ideals of $\mathbb{Z}$; we may replace the integers with the corresponding principal ideals. The formula acquires the form |
\item The preceding formula may be presented in terms of ideals of $\mathbb{Z}$; we may replace the integers with the corresponding principal ideals. The formula acquires the form |
| $$((a)+(b))((a)\cap(b)) = (a)(b).$$ |
$$((a)+(b))((a)\cap(b)) = (a)(b).$$ |
|
\item The recent formula is valid also for other than principal ideals and even in so general systems as the Pr\"ufer rings; in fact, it could be taken as defining property of these rings: Let $R$ be a commutative ring with non-zero unity.
|
\item The recent formula is valid also for other than principal ideals and even in so general systems as the Pr\"ufer rings; in fact, it could be taken as defining property of these rings. Let $R$ be a commutative ring with non-zero unity.
|
| $R$ is a Pr\"ufer ring iff the {\em Jensen's formula} |
$R$ is a Pr\"ufer ring iff the {\em Jensen's formula} |
| $$(\mathfrak{a}+\mathfrak{b})(\mathfrak{a}\cap\mathfrak{b}) = \mathfrak{ab}$$ |
$$(\mathfrak{a}+\mathfrak{b})(\mathfrak{a}\cap\mathfrak{b}) = \mathfrak{ab}$$ |
| is true for all ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, with at least one of them having non-zero-divisors. |
is true for all ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, with at least one of them having non-zero-divisors. |
| \end{enumerate} |
\end{enumerate} |
