Yes, a tooth can be infected under a bridge. A tooth that is connected to a bridge that is not cleaned properly will accumulate plaque and bacteria, which can lead to a cavity on that tooth or even both teeth that are used to connect the bridge. If this cavity progresses, it can lead to the tooth needing a root canal, which, if left untreated, will develop an infection at the end of the root. There are other reasons why a tooth can get an infection, for example, if a tooth is fractured, it can develop an infection as well.
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