# 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 $ab$ with  $a\in\mathfrak{a}$  and  $b\in\mathfrak{b}$.  It is straightforward to verify the following facts:

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

• 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 ProductOfLeftAndRightIdeal 2013-03-22 17:38:09 2013-03-22 17:38:09 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 16D25 ProductOfIdeals Intersection IdealMultiplicationLaws