Thinner tires on same rims?

Hi everyone, first time poster looking for some guidance

I'm wondering if it's ok to use thinner tires on my rims and to what extent. Right now I have stock 26x2 tires on the Matrix 550 rims they came on (can't find it's dimensions). I'm not sure to what extent I can put thinner tires on it until new rims are required.

The reasons for wanting to do this are that, primarily, I'm learning about bicycle components and so I'm screwing around with my old mountain bike. I'm going to be in the market for a road bike soon too so I want to learn what I can before buying.

Try here. Full of useful info.
Bicycle Tires from Harris Cyclery

One thing to keep in mind: 26" wheels will make a narrower tyre harsher, compared to same width on 700C. Don't get caught by the idea that narrower/harder = faster. Not always the case -- particularly on uneven surfaces like chipseal.

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