Polyimide, polyimide-based polymer electrolyte membrane, membrane-electrode assembly, and polymer electrolyte fuel cell

A polyimide comprising a structural unit (P) represented by the following formula (1):

• where
• A1 in a partial structure represented by [—O-A1-] is:
• a divalent aliphatic group R1 which has a carbon number of 1 to 10 and may have a substituent;
• a divalent aromatic group Ar1 which contains 1 to 4 rings and may have a substituent;
• a group represented by formula [—Ar2—Z1—Ar3—] (where Ar2 and Ar3, which may be the same as or different from each other, are each a divalent aromatic group which contains 1 to 4 rings and may have a substituent, and Z1 is a direct bond (−), an ether group (—O—), a thioether group (—S—), or a sulfone group (—SO2—));
• a group represented by formula [—R2—Ar4—] (where R2 is a divalent aliphatic group which has a carbon number of 1 to 10 and may have a substituent, and Ar4 is a divalent aromatic group which contains 1 to 4 rings and may have a substituent);
• or
• a group represented by formula [—Ars—R3—Ar6—] (where Ar5 and Ar6, which may be the same as or different from each other, are each a divalent aromatic group which contains 1 to 4 rings and may have a substituent, and R3 is a divalent aliphatic group which has a carbon number of 1 to 10 and may have a substituent),
• the substituent which the aliphatic groups R1, R2 and R3, and the aromatic groups Ar1, Ar2, Ar3, Ar4, Ar5 and Ar6 may have is at least one selected from a methyl group, an ethyl group, a propyl group, a butyl group, a trifluoromethyl group, a phenyl group, a phenoxy group, a phenylthio group, and a benzenesulfonyl group, and
• C1 is a tetravalent group.