I could very well be wrong, but I seem to recall that he would only need to bear the ring on his arms while his brother was alive. Once he became de jure duke (and only surviving member of the line) he would bear the arms without difference (as there could be no confusion to whom the same arms belonged). In the same way his brother's arms would have been labelled while their father was alive, but not after.
In short his arms would be the same as his brother's at that point.
One of heraldry's 'unofficial' rules, Cadency was something that could result in immense confusion if it was followed rigorously. For example, it could be possible for someone to bear exactly the same arms as his uncle (which would not be permitted), or to need a mass of symbols to display that he was the third son, of a fourth son, of a first son etc. I'm led to believe therefore that it was a convenience (in England at least), to only display marks as required and when confusion might possibly arise.
Somebody with a good knowledge of heraldry should be able to give a more concrete answer though... and I'm not that guy.