Is it possible to create own ethereum public network, and use it for my token, created on Solidity?
I want to have cryptocurrency without mining option, which can be traded on markets.
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What you need is a Token which is basically a decentral application on top of a blockchain (Ethereum, NEO, Nxt, ... )
Note that this token will not have its own blockchain, but piggy backs on another blockchain in which the smart contract is deployed on.
As an example, you can check out ERC-20 Token Standard (Ethereum)