Learn Russian: Я не люблю носить с собой мокрый коврик. - I do not like to carry a wet mat with me.

The word собой is the Instrumental case of the reflexive pronoun себя (oneself). In Russian, whenever the person you are 'with' is the same as the subject of the sentence (Я), you use the reflexive pronoun. So, literally, this translates as 'I do not like to carry with myself.' Using со мной here is grammatically possible but sounds unnatural.

Why use the verb носить here instead of нести?

Both words mean 'to carry', but носить is a multidirectional verb of motion. We use it here because the sentence describes a general habit or preference ('I do not like to carry in general'), rather than a specific, one-time trip. If you were complaining about carrying the mat right now in one direction, you would use нести.

Why do мокрый коврик look like they are in the dictionary form? Shouldn't the direct object be in the Accusative case?

They actually are in the Accusative case! Because коврик (mat) is a masculine inanimate noun (a non-living thing), its Accusative form is exactly the same as its Nominative (dictionary) form. The adjective мокрый (wet) simply follows this rule and stays the same too.

What kind of mat is a коврик?

The word коврик is a diminutive (smaller) form of the word ковёр, which means carpet or rug. It is the standard word for any small mat you can easily move around, such as a yoga mat, a bath mat, or even a computer mouse pad.