law of signs under multiplication in a ring

Lemma 1.

Let R be a ring with unity, which we denote by 1. For all x,yR:


where -x denotes the additive inverse of x in R.


Here we use the fact (-1)a=-a for all aR. First, we see that:


since, clearly, the additive inverse of -a is a itself.



where we have used several times the associativity of and the fact that (-1)x=x(-1)=-x. ∎

Title law of signs under multiplication in a ring
Canonical name LawOfSignsUnderMultiplicationInARing
Date of creation 2013-03-22 14:14:03
Last modified on 2013-03-22 14:14:03
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 10
Author alozano (2414)
Entry type Derivation
Classification msc 20-00
Classification msc 16-00
Classification msc 13-00
Synonym (-x)(-y)=xy
Related topic Ring