Advanced book on Mathematics Olympiad

2.4 Abstract Algebra 95

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Figure 16

2.4.3 Rings

Rings mimic in the abstract setting the properties of the sets of integers, polynomials, or

matrices.

Definition.A ring is a setRendowed with two operations+and·(addition and multipli-

cation) such that(R,+)is an Abelian group with identity element 0 and the multiplication

satisfies

(i) (associativity)x(yz)=(xy)zfor allx, y, z∈R, and

(ii) (distributivity)x(y+z)=xy+xzand(x+y)z=xz+yzfor allx, y, z∈R.

A ring is called commutative if the multiplication is commutative. It is said to have

identity if there exists 1∈Rsuch that 1·x=x· 1 =xfor allx∈R. An elementx∈R

is called invertible if there existsx−^1 ∈Rsuch thatxx−^1 =x−^1 x=1.

We consider two examples, the second of which appeared many years ago in the

Balkan Mathematics Competition for university students.

Example.Letxandybe elements in a ring with identity. Prove that if 1−xyis invertible,

then so is 1−yx.

Solution.Letvbe the inverse of 1−xy. Thenv( 1 −xy)=( 1 −xy)v=1; hence

vxy=xyv=v−1. We compute

( 1 +yvx)( 1 −yx)= 1 −yx+yvx−yvxyx= 1 −yx+yvx−y(v− 1 )x= 1.

A similar verification shows that( 1 −yx)( 1 +yvx)=1. It follows that 1−yxis
invertible and its inverse is 1+yvx. 

Example.Prove that if in a ringR(not necessarily with identity element)x^3 =xfor all

x∈R, then the ring is commutative.

Solution.Forx, y∈R, we have

xy^2 −y^2 xy^2 =(xy^2 −y^2 xy^2 )^3 =xy^2 xy^2 xy^2 −xy^2 xy^2 y^2 xy^2 −xy^2 y^2 xy^2 xy^2

−y^2 xy^2 xy^2 xy^2 +y^2 xy^2 xy^2 y^2 xy^2 +y^2 xy^2 y^2 xy^2 xy^2

−y^2 xy^2 y^2 xy^2 y^2 xy^2 +xy^2 y^2 xy^2 y^2 xy^2.