Posted by: holdenlee | September 13, 2010

## B7s 9/13/10

Problem 1: (Putnam and Beyond, 296) Show that if a (noncommutative) ring $R$ with identity has three elements $a,b,c$ such that

1. $ab=ba,bc=cb$
2. for any $x,y\in R$, $bx=by$ implies $x=y$
3. $ca=b$ but $ac\neq b$

then the ring cannot be finite.

Solution: Suppose the ring is finite. Let $f_r:R\to R$ denote left multiplication by $r$. By condition (2), $f_b$ is an injective function from $R$ to $R$. Since $R$ is finite, this must be a bijective function. Thus there exists $d$ such that $f_b(d)=1$, or $bd=1$, i.e. the right inverse exists; then $b^{-1}=d$ exists.

Next note by (3), $f_b=f_cf_a$. Since $f_b$ is bijective, and $f_c,f_a$ are functions $R\to R$, $f_c,f_a$ must also be bijective. Thus by the same argument as before $a^{-1},c^{-1}$ exist. Then conjugating $ca=b$ by $a$ gives $ac=acaa^{-1}=aba^{-1}=baa^{-1}=b$ using (1); this is a contradiction. Thus $R$ is infinite.

Problem 2: (Miklos Schweitzer, 1987/1) The numbers 1 to N are colored with 3 colors such that each color appears more than $\frac{N}{4}$ times. Prove that the solution $x+y=z$ has a solution with $x,y,z$ distinct colors.