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Hats There are hats available to buy in the southern empty house. When you get your first achievement a mouse will move in and you will be able to buy your first hat. Hats are purely cosmetics.
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Rings there are a few rings in the game. Magnet rings increase your pick-up radius. Golden rings increase light-radius around you. There are also rings that improve attack speed and critical chance which are buyable later in the game in the adventurer guild.

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Shoes are mostly there to increase your immunity and defense. There are various ways to get them. Either you can buy them from the Adventurer's Guild or you can find a pair in the mines.