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OK, so MA is your resident state.
The way it works is that MA will give you a credit against your MA income tax for the tax you paid to Rhode Island.
You have to file a Non-Resident return with Rhode Island and claim your RI withholding against any RI tax.
Then when you file your MA Resident return on Form 1, you will be able to claim a credit for the actual RI tax you paid on the same RI earned income that is also reported on your MA return, so that there's no double tax on the same income.
On your MA Form 1, you will fill out Schedule Z, Line 9 Worksheet - Income Tax Paid to Another Jurisdiction;
The credit amount that you compute is the smaller of the actual tax on your RI return or the amount of MA tax on the income you earned in RI. The amount of the credit from Schedule Z is entered on Line 30 of Page 2 of your MA Form 1.
Let me know if you have any questions.