You are here
Homeproduct of left and right ideal
Primary tabs
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 and $\mathfrak{b}$ a right ideal, $\mathfrak{ab}$ is a twosided ideal of $R$.

If both $\mathfrak{a}$ and $\mathfrak{b}$ are twosided ideals, then $\mathfrak{ab}\subseteq\mathfrak{a}\cap\mathfrak{b}$.
Related:
ProductOfIdeals, Intersection, IdealMultiplicationLaws
Type of Math Object:
Theorem
Major Section:
Reference
Parent:
Groups audience:
Mathematics Subject Classification
16D25 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections