Third isomorphism theorem for rings

Builds on Quotient ring Ring isomorphism

Statement

Let $R$ be a ring and let $I \subseteq J$ be two-sided ideals of $R$ . Then $J/I = \{a + I : a \in J\}$ is a two-sided ideal of the quotient ring $R/I$ . The map $(R/I)/(J/I) \to R/J$ , $(a + I) + J/I \mapsto a + J$ , is a ring isomorphism.

Remarks

The proof applies the first isomorphism theorem to the map $R/I \to R/J$ , $a + I \mapsto a + J$ . It is surjective, and its kernel is $J/I$ . For example, $6\mathbb{Z} \subseteq 2\mathbb{Z}$ , and $(\mathbb{Z}/6\mathbb{Z})/(2\mathbb{Z}/6\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$ .

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Group concept
Ring definition
Ideal definition
Ring homomorphism definition
Two-sided ideal definition
Ring isomorphism definition
Quotient ring definition
Third isomorphism theorem for rings statement

Tag	`073C80EC63`
URI	`m2://073C80EC63`
Kind	statement, theorem
Writing	W0 stub
MSC	16D25
Status	This copy of the site has no build data. Run `m2 status 073C80EC63`.
Uses	Quotient ring, Ring isomorphism

Third isomorphism theorem for rings

Statement

Remarks

Reading path

Further reading

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