Isomorphism theorems

Ring isomorphism theorems

The isomorphism theorems for rings are expressed as follows

First isomorphism theorem

Let 𝜑 :𝑅 →𝑇 be a ring homomorphism. Then the quotient by the kernel is isomorphic to the image: ring

𝑅ker⁡𝜑≅im⁡𝜑≤𝑇

Third isomorphism theorem

Let 𝐼,𝐽 ⊴𝑅 be ideals with 𝐼 ⊆𝐽. Then 𝐽/𝐼 ⊴𝑅/𝐼 and ring

𝑅/𝐼𝐽/𝐼≅𝑅𝐽

Fourth isomorphism theorem

Let 𝐼 ⊴𝑅 be an ideal. Then the map

Φ:[[𝐼,𝑅]]𝖱𝗇𝗀→[[0,𝑅/𝐼]]𝖱𝗇𝗀𝐴↦𝐴/𝑅

from subrngs containing 𝐼 to subrngs of 𝑅/𝐼 is an order-preserving bijection. Moreover, 𝐴 is an ideal iff Φ(𝐴) is.

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