'byrnan hring' - Byrnan and hring are near synonyms. M-R translate the phrase as 'the ringed corselet' (in their edition) and see it as the subject of personification (in their Guide). Brady suggests it is better to understand the phrase as 'the ringing sound of the mail-coat'.
Personification is quite frequent in Beowulf; compare the poet's admission while Beowulf is dressing that his war-mail (herebyrne) is expected to explore the deep (sceolde . . . sund cunnian).