Answer to Question #350985 in Abstract Algebra for Deq

Question

3.1. If R is a ring and a, b, c, d ∈ R, evaluate (a + b)(c + d).

3.2. Prove that if a, b ∈ R, then (a + b)² = a² + ab + ba + b² where

by x² we mean xx.

Answer 1

3.1

$(a + b)(c + d) = a(c + d) + b(c + d)$

by distributive law:

$= (ac + ad) + (bc + bd) = ac + ad + bc + bd$

3.2

$(a + b)^2 = (a + b)(a + b) = a(a + b) + b(a + b)=a^2 + ab + ba + b^2$

Note that if R is not a commutative ring $ab\ne ba$