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