I am abroad. My mother is trying to deposit my money in cash which is at her home, writing deposit check. But there needed my signature..?

Take cash to bank
Deposit to your account.
That is how it works in the US.
If a check is deposited, it just takes longer to clear and the amount must appear in the account the check is drawn on.
Your signature is only needed to withdraw from your account,or use a credit card under the same account.
Banks and credit cards charge transaction fee in some cases, and not often very much.