Let R be a ring. Let a,,an E R. (a) Prove that (a,...,an) := {i=1 Titis;...

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Let R be a ring. Let a₁,,an E R. (a) Prove that (a₁,...,an) := {i=1 Titis; | Ti, 8; € R, ti € {a₁, an}} is an ideal. (b) Prove that (a₁,,an) is the smallest ideal containing a₁,,an. (That is, if I C R is any ideal containing a₁,,an, then (a₁,...,an) C I.) (c) If R is commutative, prove that (a₁,...,an) = Ra₁ + *** + Ran. Let R be a ring. Let a₁,,an E R. (a) Prove that (a₁,...,an) := {i=1 Titis; | Ti, 8; € R, ti € {a₁, an}} is an ideal. (b) Prove that (a₁,,an) is the smallest ideal containing a₁,,an. (That is, if I C R is any ideal containing a₁,,an, then (a₁,...,an) C I.) (c) If R is commutative, prove that (a₁,...,an) = Ra₁ + *** + Ran.