Artikulu hau matematika-terminoari buruzkoa da; beste esanahietarako, ikus «Gorputz (argipena) ».
Aljebra abstraktuan
(
A
,
+
,
⋅
)
{\displaystyle (A,+,\cdot )}
A
{\displaystyle A}
A
{\displaystyle A}
+
{\displaystyle +}
⋅
{\displaystyle \cdot }
A
{\displaystyle A}
egitura aljebraikoa .
(
A
,
+
,
⋅
)
{\displaystyle (A,+,\cdot )}
A
{\displaystyle A}
+
{\displaystyle +}
A
{\displaystyle A}
∀
a
,
b
,
c
∈
A
:
(
a
+
b
)
+
c
=
a
+
(
b
+
c
)
{\displaystyle \forall a,b,c\in A:(a+b)+c=a+(b+c)}
+
{\displaystyle +}
∀
a
,
b
∈
A
:
a
+
b
=
b
+
a
{\displaystyle \forall a,b\in A:a+b=b+a}
Existitzen da
0
A
∈
A
{\displaystyle 0_{A}\in A}
∀
a
∈
A
:
a
+
0
A
=
a
{\displaystyle \forall a\in A:a+0_{A}=a}
0
A
{\displaystyle 0_{A}}
+
{\displaystyle +}
A
{\displaystyle A}
∀
a
∈
A
,
∃
a
′
∈
A
:
a
+
a
′
=
0
A
{\displaystyle \forall a\in A,\exists a'\in A:a+a'=0_{A}}
⋅
{\displaystyle \cdot }
A
{\displaystyle A}
∀
a
,
b
,
c
∈
A
:
(
a
⋅
b
)
⋅
c
=
a
⋅
(
b
⋅
c
)
{\displaystyle \forall a,b,c\in A:(a\cdot b)\cdot c=a\cdot (b\cdot c)}
⋅
{\displaystyle \cdot }
∀
a
,
b
,
c
∈
A
:
a
⋅
(
b
+
c
)
=
a
⋅
b
+
a
⋅
c
{\displaystyle \forall a,b,c\in A:a\cdot (b+c)=a\cdot b+a\cdot c}
Existitzen da
1
A
∈
A
{\displaystyle 1_{A}\in A}
∀
a
∈
A
:
a
+
0
A
=
a
{\displaystyle \forall a\in A:a+0_{A}=a}
1
A
{\displaystyle 1_{A}}
⋅
{\displaystyle \cdot }
A
−
{
0
A
}
{\displaystyle A-\{0_{A}\}}
∀
a
∈
A
,
∃
a
−
1
∈
A
:
a
⋅
a
−
1
=
1
A
{\displaystyle \forall a\in A,\exists a^{-1}\in A:a\cdot a^{-1}=1_{A}}
Propietate guzti hauek betetzen dituen multzoa eta eragiketak gorputz bat eratzen dute.
Gorputzak definitzeko era sinpleagoa existitzen da:
(
A
,
+
,
⋅
)
{\displaystyle (A,+,\cdot )}
(
A
,
+
,
⋅
)
{\displaystyle (A,+,\cdot )}
eraztun ez tribiala da.Existitzen da
1
A
∈
A
{\displaystyle 1_{A}\in A}
∀
a
∈
A
:
a
⋅
1
A
=
a
{\displaystyle \forall a\in A:a\cdot 1_{A}=a}
1
A
{\displaystyle 1_{A}}
⋅
{\displaystyle \cdot }
A
−
{
0
A
}
{\displaystyle A-\{0_{A}\}}
∀
a
∈
A
,
∃
a
−
1
∈
A
:
a
⋅
a
−
1
=
1
A
{\displaystyle \forall a\in A,\exists a^{-1}\in A:a\cdot a^{-1}=1_{A}}
(
Z
,
+
,
⋅
)
{\displaystyle (\mathbb {Z} ,+,\cdot )}
(
Q
,
+
,
⋅
)
{\displaystyle (\mathbb {Q} ,+,\cdot )}
(
i
)
{\displaystyle (i)}
(
Q
,
+
,
⋅
)
{\displaystyle (\mathbb {Q} ,+,\cdot )}
(
i
i
)
{\displaystyle (ii)}
Q
{\displaystyle \mathbb {Q} }
1
{\displaystyle 1}
∀
a
∈
A
:
a
⋅
1
=
a
{\displaystyle \forall a\in A:a\cdot 1=a}
(
i
i
i
)
{\displaystyle (iii)}
Q
{\displaystyle \mathbb {Q} }
0
{\displaystyle 0}
∀
a
∈
A
,
∃
a
−
1
∈
A
:
a
⋅
a
−
1
=
1
⟹
a
−
1
=
1
a
{\displaystyle \forall a\in A,\exists a^{-1}\in A:a\cdot a^{-1}=1\Longrightarrow a^{-1}={\frac {1}{a}}}
Beraz
(
Q
,
+
,
⋅
)
{\displaystyle (\mathbb {Q} ,+,\cdot )}
Gorputzak:
(
Q
,
+
,
⋅
)
,
(
R
,
+
,
⋅
)
,
(
C
,
+
,
⋅
)
{\displaystyle (\mathbb {Q} ,+,\cdot ),(\mathbb {R} ,+,\cdot ),(\mathbb {C} ,+,\cdot )}
Datuak: Q190109