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## 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?

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?