83(viii)IfXis an infinite set, then the family of allfinite subsets ofXforms a subring of the Boolean ringB(X).Solution.True.3.2Prove that a commutative ringRhas a unique one 1; that is, ife∈Rsatisfieser=rfor allr∈R, thene=1.Solution.Assume thater=rfor allr∈R. In particular,e1=1. On theother hand, the defining property of 1 givese1=e. Hence, 1=e.3.3LetRbe a commutative ring.(i)Prove the additive cancellation law.Solution.Absent.(ii)Prove that everya∈Rhas a unique additive inverse: ifa+b=0anda+c=0, thenb=c.Solution.Absent.(iii)Ifu∈Ris a unit, prove that its inverse is unique: ifub=1 anduc=1, thenb=c.Solution.Absent.3.4(i)Prove that subtraction inZis not an associative operation.Solution.InZ,(a−b)−c=a−(b−c)=a−b+cas long asc=0.(ii)Give an example of a commutative ringRin which subtraction isassociative.Solution.IfR=F2, subtraction is the same as addition, and so it
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