Ring isomorphism theorems

The isomorphism theorems for rings are expressed as follows

First isomorphism theorem

Let $𝜑 : 𝑅 \to 𝑇$ be a ring homomorphism. Then the quotient by the kernel is isomorphic to the image: #m/thm/ring

𝑅 k e r 𝜑 ≅ i m 𝜑 \leq 𝑇

Let $𝐼, 𝐽 ⊴ 𝑅$ be ideals with $𝐼 \subseteq 𝐽$ . Then $𝐽 / 𝐼 ⊴ 𝑅 / 𝐼$ and #m/thm/ring

𝑅 / 𝐼 𝐽 / 𝐼 ≅ 𝑅 𝐽

Let $𝐼 ⊴ 𝑅$ be an ideal. Then the map

Φ : [[𝐼, 𝑅]] 𝖱 𝗇 𝗀 \to [[0, 𝑅 / 𝐼]] 𝖱 𝗇 𝗀 𝐴 \mapsto 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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