product of left and right ideal
Let $\mathrm{\pi \x9d\x94\x9e}$ and $\mathrm{\pi \x9d\x94\x9f}$ be ideals of a ring $R$.β Denote byβ $\mathrm{\pi \x9d\x94\x9e}\beta \x81\u2019\mathrm{\pi \x9d\x94\x9f}$β the subset of $R$ formed by all finite sums of products^{} $a\beta \x81\u2019b$ withβ $a\beta \x88\x88\mathrm{\pi \x9d\x94\x9e}$β andβ $b\beta \x88\x88\mathrm{\pi \x9d\x94\x9f}$.β It is straightforward to verify the following facts:

β’
If $\mathrm{\pi \x9d\x94\x9e}$ is a left (http://planetmath.org/Ideal) and $\mathrm{\pi \x9d\x94\x9f}$ a right ideal^{}, $\mathrm{\pi \x9d\x94\x9e}\beta \x81\u2019\mathrm{\pi \x9d\x94\x9f}$β is a twosided ideal of $R$.

β’
If both $\mathrm{\pi \x9d\x94\x9e}$ and $\mathrm{\pi \x9d\x94\x9f}$ are twosided ideals, thenβ $\mathrm{\pi \x9d\x94\x9e}\beta \x81\u2019\mathrm{\pi \x9d\x94\x9f}\beta \x8a\x86\mathrm{\pi \x9d\x94\x9e}\beta \x88\copyright \mathrm{\pi \x9d\x94\x9f}$.
Title  product of left and right ideal 

Canonical name  ProductOfLeftAndRightIdeal 
Date of creation  20130322 17:38:09 
Last modified on  20130322 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 