We have a result if R is commutative, then so

Carol Valentine

Carol Valentine

Answered question


We have a result if R is commutative, then so is any quotient ring R/I, for any ideal I. If we take the contrapositive, we see that if R/I is non commutative, then R is non commutative. Is there any noncommutative ring that has a commutative quotient ring?

Answer & Explanation

Mollie Nash

Mollie Nash

Beginner2022-01-07Added 33 answers

Consider a direct sum of a commutative ring M and a non-commutative ring N, say M×N. Then the quotient ring (M×N)({0}×N) is isomorphic to M, which is commutative.
Steve Hirano

Steve Hirano

Beginner2022-01-08Added 34 answers

Let R be a noncommutative ring. Take I=R.

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