Answer to Question #294455 in Abstract Algebra for Efiii jaan

Question

If A,B C are three ideals of a ring R then show that A(BC)=(AB)C

Answer 1

Let us show that $A(BC)=(AB)C.$

Let $x\in A(BC).$ Then there exist $a\in A,\ b\in B,\ c\in C$ such that $x=a(bc).$ Since the mutiplicative operation of a ring is associative, we conclude that $x=a(bc)=(ab)c\in (AB)C.$ Therefore, $A(BC)\subset(AB)C.$

On the other hand, let $x\in (AB)C.$ Then there exist $a\in A,\ b\in B,\ c\in C$ such that $x=(ab)c.$ Since the mutiplicative operation of a ring is associative, we conclude that $x=(ab)c=a(bc)\in A(BC).$ Therefore, $(AB)C\subset A(BC).$

It follows that $A(BC)=(AB)C.$