An ideal in a ring as 'analogous' to a normal subgroup of a group, but

[matheinste]

Hello all. I am in need of a quick clarification.

A text I am reading describes an ideal in a ring as 'analogous' to a normal subgroup of a group but there appears to be a slight difference in structure in that a member of the underlying additive group from which the ideal is formed operates on a member of the ideal to produce a member of the ideal, at least that is how I read it. Am I mistaken.

Matheinste.

 

[morphism]

What do you mean by "operates?" Obviously a normal subgroup and an ideal aren't going to be exactly the same. The analogy is that ideals are kernels of ring homomorphisms and thus can be used to obtain a quotient structure of the ring, just like how normal subgroups are kernels of group homomorphisms and give rise to quotient groups.

 

[matheinste]

Thankyou morphism.

Sorry, bad terminology. I meant the binary operation ( multiplication )between two members of the ring/ideal.

I'll look up the points you have made. I think I was overlooking the point that the binary operation I was referring to is the 'multiplicative' second operation and not the primary operation of the additive group.

I am looking really for an informal pointer to the structure of an ideal and then I will be able to understand the formal definition.

Matheinste

 

[Kummer]

Given a group $$G$$ and a subgroup $$H$$ the only way to create $$G/H$$ with well-defined operations is for $$H$$ to be a normal subgroup of $$G$$.

In ring theory we have a similar situation. Given a ring $$R$$ and a subring $$N$$ to create well-defined operations for $$R/N$$ we require that $$N$$ be an ideal of $$R$$.

So it is as if it plays the role of the normal subgroup in ring theory.

-Wolfgang

 

[matt grime]

... an ideal in a ring as 'analogous' to a normal subgroup of a group but there appears to be a slight difference...

yes, and that's because 'analogous' and 'are identical' are not the same thing...

 

[matheinste]

Thanks Kummer.

The parallel between Subgroups and Ideals that you have pointed out is likely to be most helpful. I must spend a couple of hours going back to basics. I must learn to walk before I can run but I think the general idea is coming through.

Thanks Matheinste.

 

[mathwonk]

to clarify kummers post further, if H is a subgrop of G, then there is a group operation on G/H = equivalence classes of elements of g under the relation xh is equivalent to x for all h in H, such that the natural map G-->G/H taking x to its equivalence class, is a homomorphism, if and only if H is a normal subgroup.

similarly, if I is an additive subgroup of the ring R, then the group R/I has a ring structure such that R-->R/I is a ring map, if and only if I is an ideal in R.

i hope this is right. try proving them.

 

[matheinste]

Thankyou all for your comments. I now understand the structure of an ideal but there is still much more additional stuff to take in and I look forward to future help.

Matheinste.