zero times an element is zero in a ring

Lemma 1.

Let R be a ring with zero element 0 (i.e. 0 is the additive identity of R). Then for any element aR we have 0a=a0=0.

0a = (0+0)a, by definition of zero
= 0a+0a, by the distributive law

Thus 0a=0a+0a. Let b be the additive inverse of 0aR. Hence:


as claimed. The proof of a0=0 is done analogously. ∎

Title zero times an element is zero in a ring
Canonical name ZeroTimesAnElementIsZeroInARing
Date of creation 2013-03-22 14:13:57
Last modified on 2013-03-22 14:13:57
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 8
Author alozano (2414)
Entry type Theorem
Classification msc 20-00
Classification msc 16-00
Classification msc 13-00
Synonym 0a=0
Related topic 1cdotAA
Related topic AbsorbingElement