product of left and right ideal

Let $\mathfrak{a}$ and $\mathfrak{b}$ be ideals of a ring $R$ . Denote by $\mathfrak{ab}$ the subset of $R$ formed by all finite sums of products $a b$ with $a\in\mathfrak{a}$ and $b\in\mathfrak{b}$ . It is straightforward to verify the following facts:

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If $\mathfrak{a}$ is a left (http://planetmath.org/Ideal) and $\mathfrak{b}$ a right ideal, $\mathfrak{ab}$ is a two-sided ideal of $R$ .
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If both $\mathfrak{a}$ and $\mathfrak{b}$ are two-sided ideals, then $\mathfrak{ab}\subseteq\mathfrak{a}\cap\mathfrak{b}$ .

Title	product of left and right ideal
Canonical name	ProductOfLeftAndRightIdeal
Date of creation	2013-03-22 17:38:09
Last modified on	2013-03-22 17:38:09
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Theorem
Classification	msc 16D25
Related topic	ProductOfIdeals
Related topic	Intersection
Related topic	IdealMultiplicationLaws