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.


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