page1884 Rings Matthias Lorentzen...mattegrisenforlag.com
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Question
Let us show that x+ker(ϕ)⊆[x].
Given a homomrphism ϕ:A→B, we define
an equivalence relation ~ on A by: x~y⇔ϕ(x)=ϕ(y).
Let y∈x+ker(ϕ). Then ϕ(y)=
ϕ(x+z)=ϕ(x)+ϕ(z)=ϕ(x)⇔ y is in
the same equivalence class as x
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