in Set Theory & Algebra
252 views
2 votes
2 votes

Every element of some ring $(R,+,*)$ is such that $a*a=a.$ This ring

  1. is commutative  
  2. is non-commutative  
  3. may or may not be commutative  
  4. none
     
in Set Theory & Algebra
by
252 views

3 Comments

Rings aren't there in syllabus ,right ?
0
0
why may or maynot commutative?
0
0
No, its actually commutative. Was a typo in key, corrected now.
0
0

1 Answer

5 votes
5 votes
Best answer
Consider

$(x+y).(x+y) = x+y$

$\implies (x+y).x + (x+y).y = x+y$

$\implies x.x + y.x + x.y + y.y = x+y$

$\implies x + y.x + x.y + y = x + y$

$\implies x+ y + y.x + x.y = x+ y$

$\implies x.y = - y.x \to (1)$

Now, we have $a.a = a \\ \implies (a+a) . (a+a) = a+a \\ \implies a.a + a.a + a.a + a.a = a + a \\ \implies a.a + a.a = 0 \\ \implies a + a = 0 \\\implies a = -a \to (2)$

From (1) and (2),

$x.y = y.x, \forall x, y$

So, the given ring is commutative.
selected by
by
Answer:

Related questions