Question #294455

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


1
Expert's answer
2022-02-07T16:27:40-0500

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

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

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

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


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS