Is an Ideal Always a Linear Space?

[psholtz]

Is an ideal always a linear space?

I'm reading a proof, where the author is essentially saying: (1) since x is in the ideal I, and (2) since y is in the ideal I; then clearly x-y is in the ideal I.

In other words, if we have two elements belonging to the same ideal, is their linear combination always also in the ideal?

 

[micromass]

Essentially yes. But you must watch out for terminology. We use the term linear space only were the scalars form a field. If they do not form a field (but merely a ring), then we use the term module instead of linear space. So the right way of saying it is: an ideal of a ring R always forms an R-module.

 

[Landau]

Just read the definition of 'ideal'. By definition, it is (also) a subgroup. This means that if x and y are in the ideal, then so is x-y.

So the right way of saying it is: an ideal of a ring R always forms an R-module.

To add: ideals of the ring R are precisely the same thing as R-submodules of the R-module R. (All these things follow directly from the definitions.)