Showing a subset is a subring?

  • Thread starter SMA_01
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In summary, the conversation is discussing whether showing Ia is a subring, given certain conditions. The conditions include: 0 being in Ia, (a-b) being in Ia for a, b in Ia, and Ia being closed under multiplication. The conversation also mentions the need to show the distributive property of multiplication over addition and associativity of multiplication and addition, although it is debated whether it is necessary.
  • #1
SMA_01
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Showing a subset is a subring?

Homework Statement



Let R be a ring and a a fixed element in R. Let Ia={x in R l ax=0}

Homework Equations





The Attempt at a Solution



I saw these conditions in my book, but I'm not sure are these conditions sufficient in showing Ia is a subring?

(1) 0 is in Ia:

Let 0 be in R, then
a(0)=0

(2) (a-b)is in Ia, for a, b in Ia:

I'm not sure how I should start this.

(3) Ia is closed under multipication.
 
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  • #2


For 2) Let m, n be any two elements from Ia. Can you show that m+n is still in Ia? What do m and n look like? What does their sum look like?

It is similar for showing that m*n is in Ia, as well.
 
  • #3


Okay, so for 2 this is what I got:
Let m,n be in Ia. Then this means that am=0 and an=0. So,
am=an
and am-an=0
so, a(m-n)=0 therefore (m-n) is in Ia. Is this correct?
Should I follow the same flow for 3?

Thanks
 
  • #4


Looks good.
 
  • #5


Note, you also need to show the distributive property of multiplication over addition holds, and associativity of multiplication and addition.
 
  • #6


Thank you
 
  • #7


kru_ said:
Note, you also need to show the distributive property of multiplication over addition holds, and associativity of multiplication and addition.

No, you do not have to show those. Just showing the three conditions in the OP is good enough.
 
  • #8


Haha. Not according to my old abstract professor! We know they are inherited by R, but failure to show is -2 points..

sigh..
 

Related to Showing a subset is a subring?

1. What is a subring?

A subring is a subset of a ring that contains the same operations and properties as the original ring. This means that the subring must also be a ring itself.

2. How do you show that a subset is a subring?

To show that a subset is a subring, you must prove that it satisfies the three essential properties of a ring: closure under addition, closure under multiplication, and additive and multiplicative identity elements. Additionally, you must show that the subset also contains the additive and multiplicative inverses of its elements.

3. Can a subring have a different identity element than its parent ring?

No, a subring must have the same identity element as its parent ring. This is because the identity element is a defining property of a ring and must be preserved in any subset that claims to be a ring.

4. Can a subring have more elements than its parent ring?

No, a subring cannot have more elements than its parent ring. In fact, it must have fewer or the same number of elements as the parent ring. This is because a subring is a subset of the parent ring and can only contain elements that are already present in the parent ring.

5. What is the importance of showing that a subset is a subring?

Showing that a subset is a subring is important because it allows us to extend the properties and operations of a ring to a smaller subset. This can help simplify computations and proofs in certain areas of mathematics, such as abstract algebra and number theory.

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