Meaning of the $I/mI$ in a given ring A

Question

I wish to understand the meaning behind the following phrase:

"Let A be a ring and let $m$ be the maximal ideal. Let $I$ be a principal ideal. Then $I/mI$ is a one dimensional $A/m$-vector space"

So what exactly is $I/mI$ ? And how do we view it as an $A/m$ vector space. (although $A/m$ is a field).

Cheers

Answer 1

Any $A$ module annihilated by $m$ is naturally an $A/m$ module.

Such is the case for $I/mI$. And yes, $A/m$ is a field, so $I/mI$ is a vector space over $A/m$.

I'll leave it to you to show that if $I$ is principal, this is at most a $1$-dimensional $A/m$ space, since you said your main question was about what the previous stuff meant.

(Hint if you were supposed to assume $A$ is local (since you said "the maximal ideal"): $Im\neq I$ by Nakayama's Lemma.