Irish island is going after hotel (5)

I believe the answer is:

innis

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'irish island' is the definition.
I know nothing about this answer so I cannot tell whether this works.

'is going after hotel' is the wordplay.
I cannot really understand how this works, but
'is' is within the answer.
'hotel' could be 'inn' (inn is a kind of hotel) and 'inn' is found in the answer.
No letters remain.
This explanation may well be incorrect...

Can you help me to learn more?

I've seen this clue in The Irish Times.

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