Mixing
quotient ring by ideal of zero divisors
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Given a commutative ring $R$ and an element $v$ in $R$, I would like to form a quotient ring in which $v$ does not have any zero divisors. So I define an ideal $I_v$ to be the set of elements $z$ of $R$ such that $zv=0$, and then I define $R_v = R/I_v$. Is there a standard name for this construction of $I_v$ and/or $R_v$, so that I might know where to read more about it? It seems similar to the localization of the ring, but not quite the same.
abstract-algebra
asked Jan 30 at 2:31
user6013user6013
1734
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add a comment |
$begingroup$
Given a commutative ring $R$ and an element $v$ in $R$, I would like to form a quotient ring in which $v$ does not have any zero divisors. So I define an ideal $I_v$ to be the set of elements $z$ of $R$ such that $zv=0$, and then I define $R_v = R/I_v$. Is there a standard name for this construction of $I_v$ and/or $R_v$, so that I might know where to read more about it? It seems similar to the localization of the ring, but not quite the same.
abstract-algebra
asked Jan 30 at 2:31
user6013user6013
1734
$endgroup$
$begingroup$
These lecture notes discuss the quotient ring to which you are referring math.hawaii.edu/~tom/old_classes/412notes6.pdf (cf. page 5)
$endgroup$
– Victoria M
Jan 30 at 3:01
$begingroup$
Thanks for the answer. However I only saw the the general definition of a quotient ring discussed there, not the example I mentioned. Sorry if I missed it.
$endgroup$
– user6013
Jan 30 at 3:15
$begingroup$
After reading some more, I think the ideal $I_v$ is known as the "annihilator" of $v$, which is defined more generally for a subset of a module over a ring (for me the subset is the single element $v$, and the module is $R$ itself). But I still haven't seen much discussion of the quotient ring in this case, for example, if it has an interesting properties I might find useful, so any help on this would be appreciated.
$endgroup$
– user6013
Jan 31 at 18:46
add a comment |
$begingroup$
Given a commutative ring $R$ and an element $v$ in $R$, I would like to form a quotient ring in which $v$ does not have any zero divisors. So I define an ideal $I_v$ to be the set of elements $z$ of $R$ such that $zv=0$, and then I define $R_v = R/I_v$. Is there a standard name for this construction of $I_v$ and/or $R_v$, so that I might know where to read more about it? It seems similar to the localization of the ring, but not quite the same.
abstract-algebra
asked Jan 30 at 2:31
user6013user6013
1734
$endgroup$
Given a commutative ring $R$ and an element $v$ in $R$, I would like to form a quotient ring in which $v$ does not have any zero divisors. So I define an ideal $I_v$ to be the set of elements $z$ of $R$ such that $zv=0$, and then I define $R_v = R/I_v$. Is there a standard name for this construction of $I_v$ and/or $R_v$, so that I might know where to read more about it? It seems similar to the localization of the ring, but not quite the same.
abstract-algebra
abstract-algebra
asked Jan 30 at 2:31
user6013user6013
1734
asked Jan 30 at 2:31
user6013user6013
1734
asked Jan 30 at 2:31
user6013user6013
1734
asked Jan 30 at 2:31
user6013user6013
1734
asked Jan 30 at 2:31
user6013user6013
1734
1734
$begingroup$
These lecture notes discuss the quotient ring to which you are referring math.hawaii.edu/~tom/old_classes/412notes6.pdf (cf. page 5)
$endgroup$
– Victoria M
Jan 30 at 3:01
$begingroup$
Thanks for the answer. However I only saw the the general definition of a quotient ring discussed there, not the example I mentioned. Sorry if I missed it.
$endgroup$
– user6013
Jan 30 at 3:15
$begingroup$
After reading some more, I think the ideal $I_v$ is known as the "annihilator" of $v$, which is defined more generally for a subset of a module over a ring (for me the subset is the single element $v$, and the module is $R$ itself). But I still haven't seen much discussion of the quotient ring in this case, for example, if it has an interesting properties I might find useful, so any help on this would be appreciated.
$endgroup$
– user6013
Jan 31 at 18:46
add a comment |
$begingroup$
These lecture notes discuss the quotient ring to which you are referring math.hawaii.edu/~tom/old_classes/412notes6.pdf (cf. page 5)
$endgroup$
– Victoria M
Jan 30 at 3:01
$begingroup$
Thanks for the answer. However I only saw the the general definition of a quotient ring discussed there, not the example I mentioned. Sorry if I missed it.
$endgroup$
– user6013
Jan 30 at 3:15
$begingroup$
After reading some more, I think the ideal $I_v$ is known as the "annihilator" of $v$, which is defined more generally for a subset of a module over a ring (for me the subset is the single element $v$, and the module is $R$ itself). But I still haven't seen much discussion of the quotient ring in this case, for example, if it has an interesting properties I might find useful, so any help on this would be appreciated.
$endgroup$
– user6013
Jan 31 at 18:46
$begingroup$
These lecture notes discuss the quotient ring to which you are referring math.hawaii.edu/~tom/old_classes/412notes6.pdf (cf. page 5)
$endgroup$
– Victoria M
Jan 30 at 3:01
$begingroup$
These lecture notes discuss the quotient ring to which you are referring math.hawaii.edu/~tom/old_classes/412notes6.pdf (cf. page 5)
$endgroup$
– Victoria M
Jan 30 at 3:01
$begingroup$
Thanks for the answer. However I only saw the the general definition of a quotient ring discussed there, not the example I mentioned. Sorry if I missed it.
$endgroup$
– user6013
Jan 30 at 3:15
$begingroup$
Thanks for the answer. However I only saw the the general definition of a quotient ring discussed there, not the example I mentioned. Sorry if I missed it.
$endgroup$
– user6013
Jan 30 at 3:15
$begingroup$
After reading some more, I think the ideal $I_v$ is known as the "annihilator" of $v$, which is defined more generally for a subset of a module over a ring (for me the subset is the single element $v$, and the module is $R$ itself). But I still haven't seen much discussion of the quotient ring in this case, for example, if it has an interesting properties I might find useful, so any help on this would be appreciated.
$endgroup$
– user6013
Jan 31 at 18:46
$begingroup$
After reading some more, I think the ideal $I_v$ is known as the "annihilator" of $v$, which is defined more generally for a subset of a module over a ring (for me the subset is the single element $v$, and the module is $R$ itself). But I still haven't seen much discussion of the quotient ring in this case, for example, if it has an interesting properties I might find useful, so any help on this would be appreciated.
$endgroup$
– user6013
Jan 31 at 18:46
add a comment |
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$begingroup$
These lecture notes discuss the quotient ring to which you are referring math.hawaii.edu/~tom/old_classes/412notes6.pdf (cf. page 5)
$endgroup$
– Victoria M
Jan 30 at 3:01
$begingroup$
Thanks for the answer. However I only saw the the general definition of a quotient ring discussed there, not the example I mentioned. Sorry if I missed it.
$endgroup$
– user6013
Jan 30 at 3:15
$begingroup$
After reading some more, I think the ideal $I_v$ is known as the "annihilator" of $v$, which is defined more generally for a subset of a module over a ring (for me the subset is the single element $v$, and the module is $R$ itself). But I still haven't seen much discussion of the quotient ring in this case, for example, if it has an interesting properties I might find useful, so any help on this would be appreciated.
$endgroup$
– user6013
Jan 31 at 18:46