# AliReza Olfati

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 Name AliReza Olfati Member for 1 year Seen Jan 24 at 12:40 Website Location Ahvaz, Iran Age 29
Everyone thinks of changing the world, but no one thinks of changing himself. "Tolstoy"
 Apr30 awarded ● Yearling Nov28 accepted Relative extremely disconnected space Nov27 comment Ideals generated by idempotent elementsDear all, I think if $R$ is an integral domain, in the ring $M_2(R)$ your statement is trivially true. because in this case, we could characterize all of idempotents. all of idempotents have the form $0$, $I_2$ or a matrix whose entries are $a_{11}=a$, $a_{12}=b$ , $a_{21}=c$, $a_{22}=1-a$ and $a,b,c$ satisfy in the relation $bc=a-a^2$. so in this case the only nonzero ideal in $M_2(R)$ which contains an idempotent is the total ring. so in this case, your claim is true. Nov26 comment Relative extremely disconnected spaceDear Alex. Please check it more precisely. You could not change the intervals in x-axis to them. please look at the basis of $(0,0)$ and $(1,0)$. each of neighborhoods of $(0,0)$ geometrically should contains the rectangular rigion which has the fixed length equal to $\frac{1}{2}$. also each of neighborhoods of $(1,0)$ contains the rectangular rigion which has the fixed length equal to $\frac{1}{2}$. so you could not change them in your favor. Nov26 comment Idempotent elements in matrix ringHi.Let $F$ be an infinite field, then there exist infinitely many distinct pair $(I,J)$ of minimal left ideals of $M_2(F)$ such that $M_2(F)=I \oplus J$. so this shows that in this case $F$ has only two trivial idempotent, But $M_2(F)$ has infinitely many nontrivial idempotent. you could find this point at exercise [11.b] page 443 of Hubgerford. So I think you should determine the property of your ring$R$ which entries of matrix ring come from it.