There's definitely a fine line to walk. However, it's not arrogant to keep whoever you report to apprised of what you're doing, why it matters to the business, and how you're accomplishing those things. Definitely no need to be a gloryhound or grandstand, but there's a lot of middle ground between arrogance and not taking credit for what you do.

Maybe self-promotion wasn't a good choice of words on my part. Effective professional communication might be a better way to put it.