# Ideals and Linear Spaces

## Main Question or Discussion Point

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?

Related Linear and Abstract Algebra News on Phys.org
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