Exercise 1881Algebra › Rings and fieldsL1Added 3 Nov 2026

Quotient rings

#quotient ring#ideal#operations

Statement

Let $I$ be an ideal of a commutative ring $R$ . Show that $R/I$ with operations $(a+I)+(b+I)=(a+b)+I$ and $(a+I)(b+I)=ab+I$ is a commutative ring.

Hint

Well-definedness is the main point. Then check the axioms.

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Solution

Well-defined. If $a+I=a'+I$ and $b+I=b'+I$ , then $a-a'\in I$ , $b-b'\in I$ . Then $(a+b)-(a'+b')\in I$ (I is additive subgroup) and $ab-a'b'=a(b-b')+(a-a')b'\in I$ (I is an ideal). So the operations are well defined.

Ring axioms. Inherited from $R$ : associativity of addition/multiplication, distributivity, commutativity. Identity: $1+I$ . Zero: $0+I=0$ .

Conclusion. $R/I$ is a commutative ring (with unit if $R$ has one).

Quotient rings

Statement

Solution

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