Is A« B an ideal of R?&nbsp;

You don’t need to show that&nbsp;ar,ra∈A&nbsp;for all&nbsp;a∈A; nor that&nbsp;a−b∈A&nbsp;(for&nbsp;a,b∈A, I assume?). Since&nbsp;A&nbsp;is an ideal, this is given.In order to prove that&nbsp;A:B&nbsp;is an ideal, you need to show that0∈A:B&nbsp;(or alternatively&nbsp;A:B≠∅, which you have already done),for all&nbsp;x,y∈A:B, you have&nbsp;x−y∈A:B&nbsp;(or alternatively&nbsp;x+y∈A:B), andfor all&nbsp;r∈R&nbsp;and&nbsp;x∈A:B, you have&nbsp;rx∈A:B&nbsp;and&nbsp;xr∈A:B.

Step by step solution to "Is A« B an ideal of R?"

College
Algebra

nisitapradhan

Answered question

2022-07-29

Is A« B an ideal of R?

Answer & Explanation

Jeffrey Jordon

Expert2022-11-08Added 2605 answers

You don’t need to show that $a r, r a \in A$ for all $a \in A$ ; nor that $a - b \in A$ (for $a, b \in A$ , I assume?). Since $A$ is an ideal, this is given.

In order to prove that $A : B$ is an ideal, you need to show that

$0 \in A : B$ (or alternatively $A : B \neq \emptyset$ , which you have already done),
for all $x, y \in A : B$ , you have $x - y \in A : B$ (or alternatively $x + y \in A : B$ ), and
for all $r \in R$ and $x \in A : B$ , you have $r x \in A : B$ and $x r \in A : B$ .

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