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Notice removed Authoritative reference needed by user38138
Bounty Ended with Eric Wofsey's answer chosen by CommunityBot
Notice added Authoritative reference needed by user38138
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user38138
user38138

This may be simple but I can not see a way. I am looking for an uncountable ring (with 1) containing a countable maximal left ideal which is not a direct summand (as a left ideal).

This may be simple but I can not see a way. I am looking for an uncountable ring (with 1) containing a countable maximal left ideal which is not a direct summand.

This may be simple but I can not see a way. I am looking for an uncountable ring (with 1) containing a countable maximal left ideal which is not a direct summand (as a left ideal).

removed comm-alg tag since OP apparently wants to also consider the noncommutative case
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Yemon Choi
Yemon Choi
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added 5 characters in body; edited tags
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user38138
user38138

This may be simple but I can not see a way. I am looking for an uncountable ring (with 1) containing a countable maximal left ideal which is not a direct summand.

This may be simple but I can not see a way. I am looking for an uncountable ring (with 1) containing a countable maximal ideal which is not a direct summand.

This may be simple but I can not see a way. I am looking for an uncountable ring (with 1) containing a countable maximal left ideal which is not a direct summand.

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user38138
user38138