Let us show that A(BC)=(AB)C.
Let x∈A(BC). Then there exist a∈A, b∈B, c∈C such that x=a(bc). Since the mutiplicative operation of a ring is associative, we conclude that x=a(bc)=(ab)c∈(AB)C. Therefore, A(BC)⊂(AB)C.
On the other hand, let x∈(AB)C. Then there exist a∈A, b∈B, c∈C such that x=(ab)c. Since the mutiplicative operation of a ring is associative, we conclude that x=(ab)c=a(bc)∈A(BC). Therefore, (AB)C⊂A(BC).
It follows that A(BC)=(AB)C.
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