proof of product of left and right ideal

Theorem 1

Let a and b be ideals of a ring R. Denote by ab the subset of R formed by all finite sums of products ab with aa and bb. Then if a is a left and b a right idealMathworldPlanetmathPlanetmath, ab is a two-sided ideal of R. If in addition both a and b are two-sided ideals, then abab.

Proof. We must show that the difference of any two elements of 𝔞𝔟 is in 𝔞𝔟, and that 𝔞𝔟 is closed under multiplication by R. But both of these operations are linear in 𝔞𝔟; that is, if they hold for elements of the form ab,a𝔞,b𝔟, then they hold for the general element of 𝔞𝔟. So we restrict our analysis to elements ab.

Clearly if a1,a2𝔞,b1,b2𝔟, then a1b1-a2b2𝔞𝔟 by definition.

If a𝔞,b𝔟,rR, then

rab=(ra)b𝔞𝔟 since 𝔞 is a left ideal
abr=a(br)𝔞𝔟 since 𝔟 is a right ideal

and thus 𝔞𝔟 is a two-sided ideal. This proves the first statement.

If 𝔞,𝔟 are two-sided ideals, then ab𝔞 since bR; similarly, ab𝔟. This proves the second statement.

Title proof of product of left and right ideal
Canonical name ProofOfProductOfLeftAndRightIdeal
Date of creation 2013-03-22 17:41:25
Last modified on 2013-03-22 17:41:25
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Proof
Classification msc 16D25