Showing a subset is a subring?

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Homework Help Overview

The discussion revolves around determining whether a specific subset of a ring, defined by the condition \( Ia = \{ x \in R \mid ax = 0 \} \), is a subring. Participants are exploring the necessary conditions for this subset to qualify as a subring within the context of ring theory.

Discussion Character

  • Conceptual clarification, Assumption checking, Mixed

Approaches and Questions Raised

  • Participants discuss the conditions required for \( Ia \) to be a subring, including the presence of zero, closure under subtraction, and closure under multiplication. There is uncertainty about whether these conditions are sufficient. Questions arise about the specific forms of elements in \( Ia \) and how to demonstrate closure properties.

Discussion Status

Some participants have provided guidance on the necessary conditions, while others express uncertainty about the completeness of their reasoning. There is a recognition of the need to verify additional properties, such as the distributive property and associativity, although some participants argue that these may not be necessary for the proof.

Contextual Notes

There is mention of potential grading implications related to the completeness of the proof, highlighting differing expectations regarding what must be shown to satisfy the requirements of the problem.

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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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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.
 


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
 


Looks good.
 


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


Thank you
 


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.
 


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

sigh..
 

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