You should not wear a ring which has symbols on it which are obviously representing another religion. For instance, a cross which is clearly a Christian cross. Symbols are like marketing logos; they advertise the religion and also subtly link a person to it; they also identify us to others as being part of that religious group. (Someone once called the Christian cross "the most effective marketing symbol in the world.")

However, not all symbols are obviously linked to a single religion. For instance, a cross which looks like a plus sign might not necessarily be a Christian cross. Both Muslims and Jews use the 6-pointed star ("star of David"/"seal of Solomon"). Some cultures use a 5-pointed star to represent the elements of the natural world, whereas some Christians oppose it. Hitler took the swastika from Hinduism and Buddhism.

Historically, there was no specific logo or symbol for Islam, although today the crescent moon and star is used a symbol. I suppose we could do worse for a symbol! However, we also do not own the crescent moon or stars.

So, you should look at the ring and decide if the symbols are clearly identifying the ring as advertising another religion (for instance, a medieval knight ring with Crusader crosses). If it is, you should not wear it. But if it is just the case that there are crosslike mounts for the gems, or whatever, and it doesn't shout "Christian cross" to the world, then that is not a problem.