Math 2.0

Associative rings and algebras Definition

Ring homomorphism

Builds on Ring

Definition

Let RR and SS be rings. A ring homomorphism from RR to SS is a map f ⁣:R→Sf \colon R \to S with these three properties.

  1. f(a+b)=f(a)+f(b)f(a + b) = f(a) + f(b) for all a,b∈Ra, b \in R.
  2. f(ab)=f(a)f(b)f(a b) = f(a) f(b) for all a,b∈Ra, b \in R.
  3. f(1)=1f(1) = 1.

Remarks

A ring homomorphism also satisfies f(0)=0f(0) = 0 and f(−a)=−f(a)f(-a) = -f(a). For each integer n≥1n \ge 1, the map ℤ→ℤ/nℤ\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} that sends aa to its class is a ring homomorphism. The map ℤ→ℤ\mathbb{Z} \to \mathbb{Z}, a↦2aa \mapsto 2a, preserves sums but is not a ring homomorphism. A bijective ring homomorphism is a ring isomorphism.

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66 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ring_homomorphism, lead section
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