Localization of a direct productLocalization of finite direct product of ring at an elementIs every local...
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Localization of a direct product
Localization of finite direct product of ring at an elementIs every local ring the localization of some other ring?Localization at a prime and direct limitsPrime ideals in an arbitrary direct product of ringsInfinite direct product of C-M ringsIdempotent direct summands of ringsConditions for a ring to be a direct product of local ringsDirect Summands of PI Rings as Right IdealsDirect product of soclesLocalization of finite direct product of ring at an elementLocalization at prime idea of the infinite direct product of $Z/2Z$
$begingroup$
Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.
abstract-algebra ring-theory commutative-algebra noncommutative-algebra
asked Jun 24 '14 at 7:59
karparvarkarparvar
2,7001716
$endgroup$
add a comment |
$begingroup$
Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.
abstract-algebra ring-theory commutative-algebra noncommutative-algebra
asked Jun 24 '14 at 7:59
karparvarkarparvar
2,7001716
$endgroup$
add a comment |
$begingroup$
Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.
abstract-algebra ring-theory commutative-algebra noncommutative-algebra
asked Jun 24 '14 at 7:59
karparvarkarparvar
2,7001716
$endgroup$
Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.
abstract-algebra ring-theory commutative-algebra noncommutative-algebra
abstract-algebra ring-theory commutative-algebra noncommutative-algebra
asked Jun 24 '14 at 7:59
karparvarkarparvar
2,7001716
asked Jun 24 '14 at 7:59
karparvarkarparvar
2,7001716
asked Jun 24 '14 at 7:59
karparvarkarparvar
2,7001716
asked Jun 24 '14 at 7:59
karparvarkarparvar
2,7001716
asked Jun 24 '14 at 7:59
karparvarkarparvar
2,7001716
2,7001716
add a comment |
add a comment |
1 Answer
1
active
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votes
$begingroup$
First prove the following: if $S_isubset R_i$ is a multiplicative set, then $$(S_1times S_2)^{-1}(R_1times R_2)simeq S_1^{-1}R_1times S_2^{-1}R_2.$$
A maximal ideal of $R_1times R_2$ is of the form $M_1times R_2$ or $R_1times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1times R_2$ at $M_1times R_2$, consider the multiplicative set $$(R_1times R_2)setminus (M_1times R_2)=(R_1setminus M_1)times R_2,$$ use the previous isomorphism and recall that $S^{-1}R=0$ if $0in S$.
answered Jun 24 '14 at 9:46
user26857user26857
39.4k124183
$endgroup$
1
$begingroup$
can you give more detail why the isomorphism is true?
$endgroup$
– annimal
Dec 9 '14 at 15:26
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
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active
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votes
$begingroup$
First prove the following: if $S_isubset R_i$ is a multiplicative set, then $$(S_1times S_2)^{-1}(R_1times R_2)simeq S_1^{-1}R_1times S_2^{-1}R_2.$$
A maximal ideal of $R_1times R_2$ is of the form $M_1times R_2$ or $R_1times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1times R_2$ at $M_1times R_2$, consider the multiplicative set $$(R_1times R_2)setminus (M_1times R_2)=(R_1setminus M_1)times R_2,$$ use the previous isomorphism and recall that $S^{-1}R=0$ if $0in S$.
answered Jun 24 '14 at 9:46
user26857user26857
39.4k124183
$endgroup$
1
$begingroup$
can you give more detail why the isomorphism is true?
$endgroup$
– annimal
Dec 9 '14 at 15:26
add a comment |
$begingroup$
First prove the following: if $S_isubset R_i$ is a multiplicative set, then $$(S_1times S_2)^{-1}(R_1times R_2)simeq S_1^{-1}R_1times S_2^{-1}R_2.$$
A maximal ideal of $R_1times R_2$ is of the form $M_1times R_2$ or $R_1times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1times R_2$ at $M_1times R_2$, consider the multiplicative set $$(R_1times R_2)setminus (M_1times R_2)=(R_1setminus M_1)times R_2,$$ use the previous isomorphism and recall that $S^{-1}R=0$ if $0in S$.
answered Jun 24 '14 at 9:46
user26857user26857
39.4k124183
$endgroup$
1
$begingroup$
can you give more detail why the isomorphism is true?
$endgroup$
– annimal
Dec 9 '14 at 15:26
add a comment |
$begingroup$
First prove the following: if $S_isubset R_i$ is a multiplicative set, then $$(S_1times S_2)^{-1}(R_1times R_2)simeq S_1^{-1}R_1times S_2^{-1}R_2.$$
A maximal ideal of $R_1times R_2$ is of the form $M_1times R_2$ or $R_1times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1times R_2$ at $M_1times R_2$, consider the multiplicative set $$(R_1times R_2)setminus (M_1times R_2)=(R_1setminus M_1)times R_2,$$ use the previous isomorphism and recall that $S^{-1}R=0$ if $0in S$.
answered Jun 24 '14 at 9:46
user26857user26857
39.4k124183
$endgroup$
First prove the following: if $S_isubset R_i$ is a multiplicative set, then $$(S_1times S_2)^{-1}(R_1times R_2)simeq S_1^{-1}R_1times S_2^{-1}R_2.$$
A maximal ideal of $R_1times R_2$ is of the form $M_1times R_2$ or $R_1times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1times R_2$ at $M_1times R_2$, consider the multiplicative set $$(R_1times R_2)setminus (M_1times R_2)=(R_1setminus M_1)times R_2,$$ use the previous isomorphism and recall that $S^{-1}R=0$ if $0in S$.
answered Jun 24 '14 at 9:46
user26857user26857
39.4k124183
edited Mar 13 at 6:40
answered Jun 24 '14 at 9:46
user26857user26857
39.4k124183
answered Jun 24 '14 at 9:46
user26857user26857
39.4k124183
answered Jun 24 '14 at 9:46
user26857user26857
39.4k124183
39.4k124183
1
$begingroup$
can you give more detail why the isomorphism is true?
$endgroup$
– annimal
Dec 9 '14 at 15:26
add a comment |
1
$begingroup$
can you give more detail why the isomorphism is true?
$endgroup$
– annimal
Dec 9 '14 at 15:26
1
1
$begingroup$
can you give more detail why the isomorphism is true?
$endgroup$
– annimal
Dec 9 '14 at 15:26
$begingroup$
can you give more detail why the isomorphism is true?
$endgroup$
– annimal
Dec 9 '14 at 15:26
add a comment |
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