Alternate expression for definition of an Ideal?

In summary, the two statements "Ar is contained in A for all r in R" and "aR is contained in A for all a in A" define the same ideal A, which is a right ideal in the additive subgroup R.
  • #1
PsychonautQQ
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If A is an additive subgroup of a ring R, A is said to be an ideal if Ra is contained in A for all a in A; that is, if every multiple of an element of A is again in A.

Is it true that A is an ideal of R if Ar is contained in A for all r in R? To me it seems like they are equivalent statements but I'm not sure.

Update: Second thought, I believe these are NOT equivalent statements. I'm looking at the proof that explains why if 1 is in A then A = R, and you could not employ this argument under the faulty definition I tried to push. Anyone want to shed further light on the difference between these statements?
 
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  • #3
Ah right, didn't mean to switch those around. Would it be fair to say that the following statements define the same ideal A?

Ar is contained in A for all r in R
aR is contained in A for all a in A
 
  • #4
PsychonautQQ said:
Ah right, didn't mean to switch those around. Would it be fair to say that the following statements define the same ideal A?

Ar is contained in A for all r in R
aR is contained in A for all a in A
What makes you think that they are the same?
 
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  • #5
PsychonautQQ said:
Ah right, didn't mean to switch those around. Would it be fair to say that the following statements define the same ideal A?

Ar is contained in A for all r in R
aR is contained in A for all a in A
Assuming that ##A## is an additive subgroup of ##R##, I think either of these conditions implies that ##A## is a right ideal.

In either case, we need to verify that if ##a \in A## and ##r \in R##, then ##ar \in A##.

Suppose the first condition holds. If ##a \in A## and ##r \in R##, then ##ar \in Ar \subseteq A##, so ##A## is a right ideal.

Now suppose the second condition holds. If ##a \in A## and ##r \in R##, then ##ar \in aR \subseteq A##, so ##A## is a right ideal.
 
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1. What is an ideal in scientific terms?

An ideal is a theoretical concept that represents the perfect or optimal state of something. It is often used as a standard for comparison or a goal to strive for.

2. How is an ideal different from a real-life situation?

An ideal is a theoretical concept, while real-life situations are influenced by various factors and may not be perfect or optimal. Ideals are used as a reference point, but they may not always be achievable in reality.

3. Can ideals change over time?

Yes, ideals can change over time as our understanding and knowledge of a particular subject or concept evolves. What may have been considered an ideal in the past may no longer be applicable in the present.

4. Are there different types of ideals in science?

Yes, there are different types of ideals in science, such as the theoretical ideal, practical ideal, and idealized models. Each type serves a different purpose and has its own set of assumptions and limitations.

5. How are ideals used in scientific research?

Ideals are often used as a benchmark for evaluating the performance or effectiveness of a system or process. They can also serve as a starting point for developing new theories or models to explain complex phenomena.

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