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Moyes, who has received a one-match suspension, was sent off for appearing to kick the ball straight towards a ballboy with his side already 3-1 down.
Aaron Cresswell had been sent off earlier in the game and Rice - who will miss West Ham’s next to UEFA games - was seen berating match official Gil Manzano in the tunnel after the defeat.
The England international will miss both legs of their UEFA Europa Conference League play-off in August.
UEFA have also fined West Ham £1,502 for their fans’ use of fireworks at Frankfurt’s Deutsche Bank Park stadium.
The governing body also said Frankfurt would play one match behind closed doors, subject to a probationary two-year period, and fined £68,684 in total for six offences, including their fans’ pitch invasion, throwing objects and insufficient protection of the playing area against intruders.
UEFA said: "Eintracht Frankfurt - Incidents and decision: Invasion of the field of play, Art. 16(2)(a) DR, throwing of objects, Art. 16(2)(b) DR, lighting of fireworks/objects, Art. 16(2)(c) DR, provocative offensive messages, Art. 16(2)(e) DR, blocking of public passageways, Art. 38 SSR, insufficient protection of the playing area against intruders, Art. 40.01 SSR.
"One (1) match behind closed doors subject to probationary period of two (2) years. Fine: E80,000 (£68,684)."
Additional reporting from PA