factor rings

**Ryan0710** · July 31st 2008, 11:54 PM

Draw up an addition and a multiplication table for the factor ring 2

Z/8Z.What known ring is 2Z/8Z isomorphic

to?

**ThePerfectHacker** · August 1st 2008, 09:54 AM

Draw up an addition and a multiplication table for the factor ring 2
Z/8Z.What known ring is 2Z/8Z isomorphic to

If $\phi :R_1\to R_2$ is an isomorphism of commutative rings and $I_1$ is an ideal of $R_1$ then $\phi (I_1) = I_2$ is an ideal of $R_2$ and furthermore $R_1/I_1\simeq R_2/I_2$ . Define $\phi: 2\mathbb{Z}\to \mathbb{Z}$ by $\phi(x) = \tfrac{x}{2}$ . This is an isomorphism. Let $I_1 = 8\mathbb{Z}$ , this is an ideal of $2\mathbb{Z}$ . Note that $I_2 = \phi (I_1) = 4\mathbb{Z}$ . Thus, $2\mathbb{Z}/I_1 \simeq \mathbb{Z}/I_2 = \mathbb{Z}_4$ .

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