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Midfielder Rice rammed home a rebound in the 38th minute to reward Arsenal's domination against a defensively-minded West Ham who failed to register a shot on target.
Bukayo Saka then made the points safe from the penalty spot in the 67th minute to make it a satisfying 300th game in charge for Arsenal manager Mikel Arteta.
It capped a fine week for Arsenal who snatched a late comeback win at Newcastle United last weekend before beating Olympiakos in the Champions League on Wednesday.