The McCoy property in Ohm–Rush algebras
An Ohm–Rush algebra R → S is called McCoy if for any zero-divisor f in S , its content c ( f ) has nonzero annihilator in R , because McCoy proved this when S = R [ x ] . We answer a question of Nasehpour by giving a class of examples of faithfully flat Ohm–Rush algebras with the McCoy property that...
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| Veröffentlicht in: | Beiträge zur Algebra und Geometrie 2022-03, Vol.63 (1), p.209-214 |
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| container_title | Beiträge zur Algebra und Geometrie |
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| creator | Epstein, Neil |
| description | An Ohm–Rush algebra
R
→
S
is called
McCoy
if for any zero-divisor
f
in
S
, its content
c
(
f
) has nonzero annihilator in
R
, because McCoy proved this when
S
=
R
[
x
]
. We answer a question of Nasehpour by giving a class of examples of faithfully flat Ohm–Rush algebras with the McCoy property that are not weak content algebras. However, we show that a faithfully flat Ohm–Rush algebra is a weak content algebra iff
R
/
I
→
S
/
I
S
is McCoy for all radical (resp. prime) ideals
I
of
R
. When
R
is Noetherian (or has the more general
fidel (A)
property), we show that it is equivalent that
R
/
I
→
S
/
I
S
is McCoy for all ideals. |
| doi_str_mv | 10.1007/s13366-021-00562-7 |
| format | Article |
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R
→
S
is called
McCoy
if for any zero-divisor
f
in
S
, its content
c
(
f
) has nonzero annihilator in
R
, because McCoy proved this when
S
=
R
[
x
]
. We answer a question of Nasehpour by giving a class of examples of faithfully flat Ohm–Rush algebras with the McCoy property that are not weak content algebras. However, we show that a faithfully flat Ohm–Rush algebra is a weak content algebra iff
R
/
I
→
S
/
I
S
is McCoy for all radical (resp. prime) ideals
I
of
R
. When
R
is Noetherian (or has the more general
fidel (A)
property), we show that it is equivalent that
R
/
I
→
S
/
I
S
is McCoy for all ideals.</description><identifier>ISSN: 0138-4821</identifier><identifier>EISSN: 2191-0383</identifier><identifier>DOI: 10.1007/s13366-021-00562-7</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algebra ; Algebraic Geometry ; Convex and Discrete Geometry ; Geometry ; Mathematics ; Mathematics and Statistics ; Original Paper</subject><ispartof>Beiträge zur Algebra und Geometrie, 2022-03, Vol.63 (1), p.209-214</ispartof><rights>The Managing Editors 2021</rights><rights>The Managing Editors 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-d824efbe66cba1282242d49fc00620b290332e0cf5df803464c2da96994c4e23</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s13366-021-00562-7$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s13366-021-00562-7$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Epstein, Neil</creatorcontrib><title>The McCoy property in Ohm–Rush algebras</title><title>Beiträge zur Algebra und Geometrie</title><addtitle>Beitr Algebra Geom</addtitle><description>An Ohm–Rush algebra
R
→
S
is called
McCoy
if for any zero-divisor
f
in
S
, its content
c
(
f
) has nonzero annihilator in
R
, because McCoy proved this when
S
=
R
[
x
]
. We answer a question of Nasehpour by giving a class of examples of faithfully flat Ohm–Rush algebras with the McCoy property that are not weak content algebras. However, we show that a faithfully flat Ohm–Rush algebra is a weak content algebra iff
R
/
I
→
S
/
I
S
is McCoy for all radical (resp. prime) ideals
I
of
R
. When
R
is Noetherian (or has the more general
fidel (A)
property), we show that it is equivalent that
R
/
I
→
S
/
I
S
is McCoy for all ideals.</description><subject>Algebra</subject><subject>Algebraic Geometry</subject><subject>Convex and Discrete Geometry</subject><subject>Geometry</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Paper</subject><issn>0138-4821</issn><issn>2191-0383</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kE1OwzAQhS0EEqFwAVaRWLEwjMeOf5ao4k8qqoS6txLHaVq1TbCbRXfcgRtyEgxBYsdqNNL33pt5hFwyuGEA6jYyzqWkgIwCFBKpOiIZMpNWrvkxyYBxTYVGdkrOYlwDgFRKZeR60fr8xU27Q96Hrvdhf8hXu3zebj_fP16H2OblZumrUMZzctKUm-gvfueELB7uF9MnOps_Pk_vZtShgj2tNQrfVF5KV5UMNaLAWpjGpUSECg1wjh5cU9SNBi6kcFiXRhojnPDIJ-RqtE3nvA0-7u26G8IuJVqU3KQPlSgShSPlQhdj8I3tw2pbhoNlYL8bsWMjNvH2pxGrkoiPopjg3dKHP-t_VF9EZmIK</recordid><startdate>20220301</startdate><enddate>20220301</enddate><creator>Epstein, Neil</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20220301</creationdate><title>The McCoy property in Ohm–Rush algebras</title><author>Epstein, Neil</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-d824efbe66cba1282242d49fc00620b290332e0cf5df803464c2da96994c4e23</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Algebra</topic><topic>Algebraic Geometry</topic><topic>Convex and Discrete Geometry</topic><topic>Geometry</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Epstein, Neil</creatorcontrib><collection>CrossRef</collection><jtitle>Beiträge zur Algebra und Geometrie</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Epstein, Neil</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The McCoy property in Ohm–Rush algebras</atitle><jtitle>Beiträge zur Algebra und Geometrie</jtitle><stitle>Beitr Algebra Geom</stitle><date>2022-03-01</date><risdate>2022</risdate><volume>63</volume><issue>1</issue><spage>209</spage><epage>214</epage><pages>209-214</pages><issn>0138-4821</issn><eissn>2191-0383</eissn><abstract>An Ohm–Rush algebra
R
→
S
is called
McCoy
if for any zero-divisor
f
in
S
, its content
c
(
f
) has nonzero annihilator in
R
, because McCoy proved this when
S
=
R
[
x
]
. We answer a question of Nasehpour by giving a class of examples of faithfully flat Ohm–Rush algebras with the McCoy property that are not weak content algebras. However, we show that a faithfully flat Ohm–Rush algebra is a weak content algebra iff
R
/
I
→
S
/
I
S
is McCoy for all radical (resp. prime) ideals
I
of
R
. When
R
is Noetherian (or has the more general
fidel (A)
property), we show that it is equivalent that
R
/
I
→
S
/
I
S
is McCoy for all ideals.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s13366-021-00562-7</doi><tpages>6</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0138-4821 |
| ispartof | Beiträge zur Algebra und Geometrie, 2022-03, Vol.63 (1), p.209-214 |
| issn | 0138-4821 2191-0383 |
| language | eng |
| recordid | cdi_proquest_journals_2639021745 |
| source | SpringerNature Journals |
| subjects | Algebra Algebraic Geometry Convex and Discrete Geometry Geometry Mathematics Mathematics and Statistics Original Paper |
| title | The McCoy property in Ohm–Rush algebras |
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