Is corresponding with a righteous man via the Internet, for the purpose of inviting to Allah and exchanging information, considered something religiously impermissible for which one may be punished, even if the relationship is purely fraternal?

The sin you are committing must cease, for the Messenger of Allah, peace and blessings be upon him, said: "Sin is that which wavers in your breast and which you dislike people to know about." The basic principle is for a woman to remain in her home and avoid mixing with men, except for a necessity such as buying and selling, seeking a fatwa (definition), or filing a complaint, provided there is no fear of temptation. It is permissible for her to write on the internet to disseminate knowledge or good, briefly and concisely, and to avoid phrases of praise and commendation. As for private conversations, they are not permissible.