On (m, n)-absorbing prime ideals and (m, n)-absorbing ideals of commutative rings
Let R be a commutative ring with nonzero identity. In this paper, we introduce and investigate a generalization of 1-absorbing prime ideals. Let m , n be nonzero positive integers such that m > n . A proper ideal I of R is said to be an ( m , n )-absorbing prime ideal if whenever nonunit elemen...
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| Veröffentlicht in: | São Paulo Journal of Mathematical Sciences 2023-12, Vol.17 (2), p.888-901 |
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| creator | Badawi, Ayman El Khalfi, Abdelhaq Mahdou, Najib |
| description | Let
R
be a commutative ring with nonzero identity. In this paper, we introduce and investigate a generalization of 1-absorbing prime ideals. Let
m
,
n
be nonzero positive integers such that
m
>
n
. A proper ideal
I
of
R
is said to be an (
m
,
n
)-absorbing prime ideal if whenever nonunit elements
a
1
,
.
.
.
,
a
m
∈
R
and
a
1
.
.
.
a
m
∈
I
, then
a
1
.
.
.
a
n
∈
I
or
a
n
+
1
.
.
.
a
m
∈
I
.
We give some basic properties of this class of ideals and we study (
m
,
n
)-absorbing prime ideals of localization of rings, direct product of rings and trivial ring extensions. A proper ideal
I
of
R
is called an
AB
-(
m
,
n
)-absorbing ideal of
R
if whenever
a
1
⋯
a
m
∈
I
for some elements
a
1
,
.
.
.
,
a
m
∈
R
, then there are
n
of the
a
i
’s whose product is in
I
. A proper ideal
I
of
R
is called an (
m
,
n
)-absorbing ideal of
R
if whenever
a
1
⋯
a
m
∈
I
for some nonunit elements
a
1
,
.
.
.
,
a
m
∈
R
, then there are
n
of the
a
i
’s whose product is in
I
. We study some connections between (
m
,
n
)-absorbing prime ideals, (
m
,
n
)-absorbing ideals and
AB
-(
m
,
n
)-absorbing ideals of commutative rings. |
| doi_str_mv | 10.1007/s40863-022-00349-1 |
| format | Article |
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R
be a commutative ring with nonzero identity. In this paper, we introduce and investigate a generalization of 1-absorbing prime ideals. Let
m
,
n
be nonzero positive integers such that
m
>
n
. A proper ideal
I
of
R
is said to be an (
m
,
n
)-absorbing prime ideal if whenever nonunit elements
a
1
,
.
.
.
,
a
m
∈
R
and
a
1
.
.
.
a
m
∈
I
, then
a
1
.
.
.
a
n
∈
I
or
a
n
+
1
.
.
.
a
m
∈
I
.
We give some basic properties of this class of ideals and we study (
m
,
n
)-absorbing prime ideals of localization of rings, direct product of rings and trivial ring extensions. A proper ideal
I
of
R
is called an
AB
-(
m
,
n
)-absorbing ideal of
R
if whenever
a
1
⋯
a
m
∈
I
for some elements
a
1
,
.
.
.
,
a
m
∈
R
, then there are
n
of the
a
i
’s whose product is in
I
. A proper ideal
I
of
R
is called an (
m
,
n
)-absorbing ideal of
R
if whenever
a
1
⋯
a
m
∈
I
for some nonunit elements
a
1
,
.
.
.
,
a
m
∈
R
, then there are
n
of the
a
i
’s whose product is in
I
. We study some connections between (
m
,
n
)-absorbing prime ideals, (
m
,
n
)-absorbing ideals and
AB
-(
m
,
n
)-absorbing ideals of commutative rings.</description><identifier>ISSN: 1982-6907</identifier><identifier>EISSN: 2316-9028</identifier><identifier>EISSN: 2306-9028</identifier><identifier>DOI: 10.1007/s40863-022-00349-1</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Mathematics ; Mathematics and Statistics ; Original Article ; Rings (mathematics)</subject><ispartof>São Paulo Journal of Mathematical Sciences, 2023-12, Vol.17 (2), p.888-901</ispartof><rights>Instituto de Matemática e Estatística da Universidade de São Paulo 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-23cf80ee0c80d3cfc36a80e84a8ee9894bf8eb10fbbc627540c21667978929eb3</citedby><cites>FETCH-LOGICAL-c319t-23cf80ee0c80d3cfc36a80e84a8ee9894bf8eb10fbbc627540c21667978929eb3</cites><orcidid>0000-0002-0897-8762</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s40863-022-00349-1$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s40863-022-00349-1$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Badawi, Ayman</creatorcontrib><creatorcontrib>El Khalfi, Abdelhaq</creatorcontrib><creatorcontrib>Mahdou, Najib</creatorcontrib><title>On (m, n)-absorbing prime ideals and (m, n)-absorbing ideals of commutative rings</title><title>São Paulo Journal of Mathematical Sciences</title><addtitle>São Paulo J. Math. Sci</addtitle><description>Let
R
be a commutative ring with nonzero identity. In this paper, we introduce and investigate a generalization of 1-absorbing prime ideals. Let
m
,
n
be nonzero positive integers such that
m
>
n
. A proper ideal
I
of
R
is said to be an (
m
,
n
)-absorbing prime ideal if whenever nonunit elements
a
1
,
.
.
.
,
a
m
∈
R
and
a
1
.
.
.
a
m
∈
I
, then
a
1
.
.
.
a
n
∈
I
or
a
n
+
1
.
.
.
a
m
∈
I
.
We give some basic properties of this class of ideals and we study (
m
,
n
)-absorbing prime ideals of localization of rings, direct product of rings and trivial ring extensions. A proper ideal
I
of
R
is called an
AB
-(
m
,
n
)-absorbing ideal of
R
if whenever
a
1
⋯
a
m
∈
I
for some elements
a
1
,
.
.
.
,
a
m
∈
R
, then there are
n
of the
a
i
’s whose product is in
I
. A proper ideal
I
of
R
is called an (
m
,
n
)-absorbing ideal of
R
if whenever
a
1
⋯
a
m
∈
I
for some nonunit elements
a
1
,
.
.
.
,
a
m
∈
R
, then there are
n
of the
a
i
’s whose product is in
I
. We study some connections between (
m
,
n
)-absorbing prime ideals, (
m
,
n
)-absorbing ideals and
AB
-(
m
,
n
)-absorbing ideals of commutative rings.</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Article</subject><subject>Rings (mathematics)</subject><issn>1982-6907</issn><issn>2316-9028</issn><issn>2306-9028</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNp9UM1KAzEQDqJgqX0BTwteFIxOfjabOUrxD4Qe1HNI0qxscXdrshV8G5_FJzN1C17EucwM388wHyHHDC4YQHWZJGglKHBOAYREyvbIhAumKALX-2TCUHOqEKpDMktpBblKWWEJE_K46IrT9vzrszuj1qU-uqZ7KdaxaUPRLIN9TYXtln9QdmBfF75v281gh-Y9FDFD6Ygc1BkLs12fkueb66f5HX1Y3N7Prx6oFwwHyoWvNYQAXsMyz14om3ctrQ4BNUpX6-AY1M55xatSgudMqQorjRyDE1NyMvquY_-2CWkwq34Tu3zScI0oARWWmcVHlo99SjHUZvudjR-GgdnmZ8b8TM7P_ORnWBaJUZTW25dC_LX-R_UNsQZy3Q</recordid><startdate>20231201</startdate><enddate>20231201</enddate><creator>Badawi, Ayman</creator><creator>El Khalfi, Abdelhaq</creator><creator>Mahdou, Najib</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-0897-8762</orcidid></search><sort><creationdate>20231201</creationdate><title>On (m, n)-absorbing prime ideals and (m, n)-absorbing ideals of commutative rings</title><author>Badawi, Ayman ; El Khalfi, Abdelhaq ; Mahdou, Najib</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-23cf80ee0c80d3cfc36a80e84a8ee9894bf8eb10fbbc627540c21667978929eb3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Article</topic><topic>Rings (mathematics)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Badawi, Ayman</creatorcontrib><creatorcontrib>El Khalfi, Abdelhaq</creatorcontrib><creatorcontrib>Mahdou, Najib</creatorcontrib><collection>CrossRef</collection><jtitle>São Paulo Journal of Mathematical Sciences</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Badawi, Ayman</au><au>El Khalfi, Abdelhaq</au><au>Mahdou, Najib</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On (m, n)-absorbing prime ideals and (m, n)-absorbing ideals of commutative rings</atitle><jtitle>São Paulo Journal of Mathematical Sciences</jtitle><stitle>São Paulo J. Math. Sci</stitle><date>2023-12-01</date><risdate>2023</risdate><volume>17</volume><issue>2</issue><spage>888</spage><epage>901</epage><pages>888-901</pages><issn>1982-6907</issn><eissn>2316-9028</eissn><eissn>2306-9028</eissn><abstract>Let
R
be a commutative ring with nonzero identity. In this paper, we introduce and investigate a generalization of 1-absorbing prime ideals. Let
m
,
n
be nonzero positive integers such that
m
>
n
. A proper ideal
I
of
R
is said to be an (
m
,
n
)-absorbing prime ideal if whenever nonunit elements
a
1
,
.
.
.
,
a
m
∈
R
and
a
1
.
.
.
a
m
∈
I
, then
a
1
.
.
.
a
n
∈
I
or
a
n
+
1
.
.
.
a
m
∈
I
.
We give some basic properties of this class of ideals and we study (
m
,
n
)-absorbing prime ideals of localization of rings, direct product of rings and trivial ring extensions. A proper ideal
I
of
R
is called an
AB
-(
m
,
n
)-absorbing ideal of
R
if whenever
a
1
⋯
a
m
∈
I
for some elements
a
1
,
.
.
.
,
a
m
∈
R
, then there are
n
of the
a
i
’s whose product is in
I
. A proper ideal
I
of
R
is called an (
m
,
n
)-absorbing ideal of
R
if whenever
a
1
⋯
a
m
∈
I
for some nonunit elements
a
1
,
.
.
.
,
a
m
∈
R
, then there are
n
of the
a
i
’s whose product is in
I
. We study some connections between (
m
,
n
)-absorbing prime ideals, (
m
,
n
)-absorbing ideals and
AB
-(
m
,
n
)-absorbing ideals of commutative rings.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s40863-022-00349-1</doi><tpages>14</tpages><orcidid>https://orcid.org/0000-0002-0897-8762</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1982-6907 |
| ispartof | São Paulo Journal of Mathematical Sciences, 2023-12, Vol.17 (2), p.888-901 |
| issn | 1982-6907 2316-9028 2306-9028 |
| language | eng |
| recordid | cdi_proquest_journals_2899409695 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Mathematics Mathematics and Statistics Original Article Rings (mathematics) |
| title | On (m, n)-absorbing prime ideals and (m, n)-absorbing ideals of commutative rings |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-03T17%3A28%3A13IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=On%20(m,%C2%A0n)-absorbing%20prime%20ideals%20and%20(m,%C2%A0n)-absorbing%20ideals%20of%20commutative%20rings&rft.jtitle=S%C3%A3o%20Paulo%20Journal%20of%20Mathematical%20Sciences&rft.au=Badawi,%20Ayman&rft.date=2023-12-01&rft.volume=17&rft.issue=2&rft.spage=888&rft.epage=901&rft.pages=888-901&rft.issn=1982-6907&rft.eissn=2316-9028&rft_id=info:doi/10.1007/s40863-022-00349-1&rft_dat=%3Cproquest_cross%3E2899409695%3C/proquest_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2899409695&rft_id=info:pmid/&rfr_iscdi=true |