Let 𝚁 and 𝚂 be rings, and let φ : 𝚁 → 𝚂 be a ring homomorphism. Then:

The kernel of φ is an ideal of 𝚁 ,
The image of φ is a subring of 𝚂 , and
The image of φ is isomorphic to the quotient ring 𝚁 / 𝚔𝚎𝚛 (φ) .

In particular, if φ is surjective then 𝚂 is isomorphic to 𝚁 / 𝚔𝚎𝚛 (φ) .

Let 𝚁 be a ring. Let 𝚂 be a subring of 𝚁 , and let I be an ideal of 𝚁 . Then:

The sum 𝚂 + 𝙸 = {𝚜 + 𝚒 | 𝚜 ∈ 𝚂, 𝚒 ∈ 𝙸 } is a subring of 𝚁 ,
The intersection 𝚂 ∩ 𝙸 is an ideal of 𝚂 , and
The quotient rings (𝚂 + 𝙸) / 𝙸 and 𝚂 / (𝚂 ∩ 𝙸) are isomorphic.

Let 𝚁 be a ring, and 𝙸 an ideal of 𝚁 . Then

If 𝙰 is a subring of 𝚁 such that 𝙸 ⊆ 𝙰 ⊆ 𝚁 , then 𝙰 / 𝙸 is a subring of 𝚁 / 𝙸 .
Every subring of 𝚁 / 𝙸 is of the form 𝙰 / 𝙸 for some subring 𝙰 of 𝚁 such that 𝙸 ⊆ 𝙰 ⊆ 𝚁 .
If 𝙹 is an ideal of 𝚁 such that 𝙸 ⊆ 𝙹 ⊆ 𝚁 , then 𝙹 / 𝙸 is an ideal of 𝚁 / 𝙸 .
Every ideal of 𝚁 / 𝙸 is of the form 𝙹 / 𝙸 for some ideal 𝙹 of 𝚁 such that 𝙸 ⊆ 𝙹 ⊆ 𝚁 .
If 𝙹 is an ideal of 𝚁 such that 𝙸 ⊆ 𝙹 ⊆ 𝚁 , then the quotient ring ( 𝚁 / 𝙸 ) / ( 𝙹 / 𝙸 ) .

Let 𝙸 be an ideal of 𝚁 . The correspondence 𝙰 ↔ 𝙰 / 𝙸 is an inclusion-preserving bijection between the set of subrings 𝙰 of 𝚁 that contain 𝙸 and the set of subrings of 𝚁 / 𝙸. Furthermore, 𝙰 (a subring containing 𝙸 ) is an ideal of 𝚁 if and only if 𝙰 / 𝙸 is an ideal of 𝚁 / 𝙸 .