De Wikipedia, la enciclopedia libre
Sean dos anillos
(
R
,
+
,
⋅
)
{\displaystyle (R,+,\cdot )}
(
S
,
+
,
⋅
)
{\displaystyle (S,+,\cdot )}
R
{\displaystyle R}
subanillo de
S
{\displaystyle S}
α
∈
L
{\displaystyle \alpha \in L}
Construimos la aplicación
β
:
R
[
x
]
⟶
S
{\displaystyle \beta :R[x]\longrightarrow S}
p
(
x
)
∈
R
[
x
]
{\displaystyle p(x)\in R[x]}
α
{\displaystyle \alpha }
i.e. ,
β
(
p
)
=
p
(
α
)
{\displaystyle \beta (p)=p(\alpha )}
isomorfismo de anillos (que se denomina homomorfismo evaluación ):
β
(
p
+
q
)
=
(
p
+
q
)
(
α
)
=
p
(
α
)
+
q
(
α
)
=
β
(
p
)
+
β
(
q
)
{\displaystyle \beta (p+q)=(p+q)(\alpha )=p(\alpha )+q(\alpha )=\beta (p)+\beta (q)}
β
(
p
⋅
q
)
=
(
p
⋅
q
)
(
α
)
=
p
(
α
)
⋅
q
(
α
)
=
β
(
p
)
⋅
β
(
q
)
{\displaystyle \beta (p\cdot q)=(p\cdot q)(\alpha )=p(\alpha )\cdot q(\alpha )=\beta (p)\cdot \beta (q)}
cualesquiera que sean
p
,
q
∈
R
[
x
]
{\displaystyle p,q\in R[x]}
Además, si R y S fuesen anillos y unitarios entonces:
β
(
1
)
=
1
{\displaystyle \beta (1)=1}
con lo que
β
{\displaystyle \beta }
![{\displaystyle \beta :R[x]\longrightarrow S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4674bd76f3ac1762aa0617701eda3b36c0853dbf)
![{\displaystyle p(x)\in R[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1d267fbdc25900b5a671a1858d2e2a71db5600)
![{\displaystyle p,q\in R[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cbd5f5420a7b683b219d941de2e65b538d9e0f3)
Datos: Q5901419