Ideals generated and their inclusions
Let $R=2mathbb{Z}$, then $R$ is a ring without identity, and we consider the ideal generated by $4$, that is $$(4)={4(2j)+4overline{k};|;j,overline{k}inmathbb{Z}}.$$
I must prove that $(4)=4mathbb{Z}$. Now, I proved that $(4)subseteq4mathbb{Z}$, in fact, let $xin(4)$, then $x=4(2j)+4overline{k}=4(2j+overline{k})in 4mathbb{Z}$.
Can I prove that $4mathbb{Z}subseteq (4)$? Thanks!
abstract-algebra proof-verification proof-writing proof-explanation
asked Dec 2 at 15:58
Jack J.
4431419
add a comment |
Let $R=2mathbb{Z}$, then $R$ is a ring without identity, and we consider the ideal generated by $4$, that is $$(4)={4(2j)+4overline{k};|;j,overline{k}inmathbb{Z}}.$$
I must prove that $(4)=4mathbb{Z}$. Now, I proved that $(4)subseteq4mathbb{Z}$, in fact, let $xin(4)$, then $x=4(2j)+4overline{k}=4(2j+overline{k})in 4mathbb{Z}$.
Can I prove that $4mathbb{Z}subseteq (4)$? Thanks!
abstract-algebra proof-verification proof-writing proof-explanation
asked Dec 2 at 15:58
Jack J.
4431419
add a comment |
Let $R=2mathbb{Z}$, then $R$ is a ring without identity, and we consider the ideal generated by $4$, that is $$(4)={4(2j)+4overline{k};|;j,overline{k}inmathbb{Z}}.$$
I must prove that $(4)=4mathbb{Z}$. Now, I proved that $(4)subseteq4mathbb{Z}$, in fact, let $xin(4)$, then $x=4(2j)+4overline{k}=4(2j+overline{k})in 4mathbb{Z}$.
Can I prove that $4mathbb{Z}subseteq (4)$? Thanks!
abstract-algebra proof-verification proof-writing proof-explanation
asked Dec 2 at 15:58
Jack J.
4431419
Let $R=2mathbb{Z}$, then $R$ is a ring without identity, and we consider the ideal generated by $4$, that is $$(4)={4(2j)+4overline{k};|;j,overline{k}inmathbb{Z}}.$$
I must prove that $(4)=4mathbb{Z}$. Now, I proved that $(4)subseteq4mathbb{Z}$, in fact, let $xin(4)$, then $x=4(2j)+4overline{k}=4(2j+overline{k})in 4mathbb{Z}$.
Can I prove that $4mathbb{Z}subseteq (4)$? Thanks!
abstract-algebra proof-verification proof-writing proof-explanation
abstract-algebra proof-verification proof-writing proof-explanation
asked Dec 2 at 15:58
Jack J.
4431419
asked Dec 2 at 15:58
Jack J.
4431419
asked Dec 2 at 15:58
Jack J.
4431419
asked Dec 2 at 15:58
Jack J.
4431419
asked Dec 2 at 15:58
Jack J.
4431419
4431419
add a comment |
add a comment |
1 Answer
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Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then
$$x=4k=4(2cdot0)+4kin(4)$$
answered Dec 2 at 16:13
egreg
178k1484201
add a comment |
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1 Answer
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Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then
$$x=4k=4(2cdot0)+4kin(4)$$
answered Dec 2 at 16:13
egreg
178k1484201
add a comment |
Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then
$$x=4k=4(2cdot0)+4kin(4)$$
answered Dec 2 at 16:13
egreg
178k1484201
add a comment |
Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then
$$x=4k=4(2cdot0)+4kin(4)$$
answered Dec 2 at 16:13
egreg
178k1484201
Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then
$$x=4k=4(2cdot0)+4kin(4)$$
answered Dec 2 at 16:13
egreg
178k1484201
answered Dec 2 at 16:13
egreg
178k1484201
answered Dec 2 at 16:13
egreg
178k1484201
answered Dec 2 at 16:13
egreg
178k1484201
178k1484201
add a comment |
add a comment |
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