What will be the smallest ring containing two rings?
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$begingroup$
Let $A$ be a commutative ring with $1$. Suppose $R$ and $S$ are two subrings of $A$ containing the same multiplicative unity. Then what is the description of the smallest ring containing $R$ and $S$ inside $A$ ?
In addition if $R$ and $S$ are domains can we say that the smallest ring containing them is also a domain?
I need some explanations to this. Thank you.
abstract-algebra ring-theory commutative-algebra integral-domain
asked Jan 3 at 10:50
user371231user371231
771511
$endgroup$
add a comment |
$begingroup$
Let $A$ be a commutative ring with $1$. Suppose $R$ and $S$ are two subrings of $A$ containing the same multiplicative unity. Then what is the description of the smallest ring containing $R$ and $S$ inside $A$ ?
In addition if $R$ and $S$ are domains can we say that the smallest ring containing them is also a domain?
I need some explanations to this. Thank you.
abstract-algebra ring-theory commutative-algebra integral-domain
asked Jan 3 at 10:50
user371231user371231
771511
$endgroup$
add a comment |
$begingroup$
Let $A$ be a commutative ring with $1$. Suppose $R$ and $S$ are two subrings of $A$ containing the same multiplicative unity. Then what is the description of the smallest ring containing $R$ and $S$ inside $A$ ?
In addition if $R$ and $S$ are domains can we say that the smallest ring containing them is also a domain?
I need some explanations to this. Thank you.
abstract-algebra ring-theory commutative-algebra integral-domain
asked Jan 3 at 10:50
user371231user371231
771511
$endgroup$
Let $A$ be a commutative ring with $1$. Suppose $R$ and $S$ are two subrings of $A$ containing the same multiplicative unity. Then what is the description of the smallest ring containing $R$ and $S$ inside $A$ ?
In addition if $R$ and $S$ are domains can we say that the smallest ring containing them is also a domain?
I need some explanations to this. Thank you.
abstract-algebra ring-theory commutative-algebra integral-domain
abstract-algebra ring-theory commutative-algebra integral-domain
asked Jan 3 at 10:50
user371231user371231
771511
asked Jan 3 at 10:50
user371231user371231
771511
edited Jan 3 at 14:34
user593746
asked Jan 3 at 10:50
user371231user371231
771511
asked Jan 3 at 10:50
user371231user371231
771511
asked Jan 3 at 10:50
user371231user371231
771511
771511
add a comment |
add a comment |
1 Answer
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$begingroup$
The smallest ring containing both $R$ and $S$ is the set
$$ leftsum_i=1^n r_is_i Bigg. $$
Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=mathbbZ[x,y]/(xy)$, $R= mathbbZ[x]$ and $S = mathbbZ[y]$.
answered Jan 3 at 10:59
Pierre-Guy PlamondonPierre-Guy Plamondon
8,79511639
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1 Answer
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1 Answer
1
active
oldest
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oldest
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active
oldest
votes
$begingroup$
The smallest ring containing both $R$ and $S$ is the set
$$ leftsum_i=1^n r_is_i Bigg. $$
Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=mathbbZ[x,y]/(xy)$, $R= mathbbZ[x]$ and $S = mathbbZ[y]$.
answered Jan 3 at 10:59
Pierre-Guy PlamondonPierre-Guy Plamondon
8,79511639
$endgroup$
add a comment |
$begingroup$
The smallest ring containing both $R$ and $S$ is the set
$$ leftsum_i=1^n r_is_i Bigg. $$
Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=mathbbZ[x,y]/(xy)$, $R= mathbbZ[x]$ and $S = mathbbZ[y]$.
answered Jan 3 at 10:59
Pierre-Guy PlamondonPierre-Guy Plamondon
8,79511639
$endgroup$
add a comment |
$begingroup$
The smallest ring containing both $R$ and $S$ is the set
$$ leftsum_i=1^n r_is_i Bigg. $$
Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=mathbbZ[x,y]/(xy)$, $R= mathbbZ[x]$ and $S = mathbbZ[y]$.
answered Jan 3 at 10:59
Pierre-Guy PlamondonPierre-Guy Plamondon
8,79511639
$endgroup$
The smallest ring containing both $R$ and $S$ is the set
$$ leftsum_i=1^n r_is_i Bigg. $$
Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=mathbbZ[x,y]/(xy)$, $R= mathbbZ[x]$ and $S = mathbbZ[y]$.
answered Jan 3 at 10:59
Pierre-Guy PlamondonPierre-Guy Plamondon
8,79511639
edited Jan 3 at 14:35
user593746
answered Jan 3 at 10:59
Pierre-Guy PlamondonPierre-Guy Plamondon
8,79511639
answered Jan 3 at 10:59
Pierre-Guy PlamondonPierre-Guy Plamondon
8,79511639
answered Jan 3 at 10:59
Pierre-Guy PlamondonPierre-Guy Plamondon
8,79511639
8,79511639
add a comment |
add a comment |
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