boolean ring .commutative ring

Question

Let $(R,+, . )$ be a (not necessarily commutative) ring for which $r+r=r$ for all $r \in R$. ( boolean ring ).

☆ by considering $(r+r). (r+r)$ show that $r+r=0_R$?

☆show that $(R,+,.)$ Must be a commutative ring? Using $(r+s)(r+s)$

That is a strange definition. Usually a Boolean ring is one that satisfies $r\cdot r=r$ for all $r\in R$. The only ring that satisfies $r+r=r$ for all $r$ is the zero ring. — hmakholm left over Monica, Dec 27 '14 at 18:12
Oh sorry henning you are right I miss typed it so it's r.r=r — M Alrantisi, Dec 27 '14 at 20:29

score 2 · Accepted Answer · answered Dec 27 '14 at 18:47

2

Assuming $r^2 = r, \forall r \in R$ as pointed out by Henning Makholm

(1). $(r + r)^2 = (r + r) \Rightarrow r + r + r +r = r+r \Rightarrow r + r = 0_R.$

(2). $(r+s)(r+s) = (r+s) \Rightarrow rs + sr =0_R = rs + rs \Rightarrow sr = rs.$

answered Dec 27 '14 at 18:47

Krish

1 Answers1