Arnautovic pushed for a move from Stoke and it is questionable whether West Ham was a significant step up for the Austrian forward.
Stoke have only won four games all season and they are having their worst start to a campaign since winning promotion in 2008.
Stoke City sit 15th on the table and three points from the relegation zone and face lowly placed West Ham in a crucial fixture.
The feeling around Stoke is very different at present, after Tuesday's 1-0 loss to Burnley at Turf Moor made it one win in their last seven top-flight encounters. Stoke's biggest problem this season has been the club's inability to defend. If Mark Hughes cannot improve his club's defense, he could be out of a job before Christmas.
After shocking Chelsea on the weekend, West Ham were able to get a point off of Arsenal, after a scoreless draw.
Who will come out on top in the battle of the managers Mark Hughes or David Moyes?
After the victory, boss Moyes revealed he was annoyed with his star, who missed two glorious opportunities earlier in the win, however was pleased with his side's overall performance. Only four teams have scored fewer goals than West Ham's 14. Live streaming is available on NBC Sports Gold, which you can access with a subscription.
It is in the latter position where Rice may be required in the coming weeks, depending on the severity of Noble's injury, but on Saturday's evidence he will have no issue stepping in for the West Ham skipper.
The 18-year-old has played 11 times for West Ham this season and he looks equally comfortable as a centre-back or a holding midfielder.