A TORSION-FREE ABELIAN GROUP EXISTS WHOSE QUOTIENT GROUP MODULO THE SQUARE SUBGROUP IS NOT A NIL-GROUP
The first example of a torsion-free abelian group $(A,+,0)$ such that the quotient group of $A$ modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a sub...
Saved in:
| Published in | Bulletin of the Australian Mathematical Society Vol. 94; no. 3; pp. 449 - 456 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.12.2016
|
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | The first example of a torsion-free abelian group
$(A,+,0)$
such that the quotient group of
$A$
modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen
27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from
$2$
to
$2^{\aleph _{0}}$
is presented. Ring multiplications on
$p$
-pure subgroups of the additive group of the ring of
$p$
-adic integers are investigated using only elementary methods. |
|---|---|
| AbstractList | The first example of a torsion-free abelian group
$(A,+,0)$
such that the quotient group of
$A$
modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen
27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from
$2$
to
$2^{\aleph _{0}}$
is presented. Ring multiplications on
$p$
-pure subgroups of the additive group of the ring of
$p$
-adic integers are investigated using only elementary methods. The first example of a torsion-free abelian group $(A,+,0)$ such that the quotient group of $A$ modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen 27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from $2$ to $2^{\aleph _{0}}$ is presented. Ring multiplications on $p$ -pure subgroups of the additive group of the ring of $p$ -adic integers are investigated using only elementary methods. |
| Author | WORONOWICZ, M. ANDRUSZKIEWICZ, R. R. |
| Author_xml | – sequence: 1 givenname: R. R. surname: ANDRUSZKIEWICZ fullname: ANDRUSZKIEWICZ, R. R. email: randrusz@math.uwb.edu.pl organization: Institute of Mathematics, University of Białystok, 15-267 Białystok, K. Ciołkowskiego 1M, Poland email randrusz@math.uwb.edu.pl – sequence: 2 givenname: M. surname: WORONOWICZ fullname: WORONOWICZ, M. email: mworonowicz@math.uwb.edu.pl organization: Institute of Mathematics, University of Białystok, 15-267 Białystok, K. Ciołkowskiego 1M, Poland email mworonowicz@math.uwb.edu.pl |
| BookMark | eNp1UMtOwkAUnRhMBPQD3E3iujqPTqddFhygSe1IH9FdM21nCEQotrDwb_wWv8xWSFwYV_fe87rJGYHBrt5pAG4xuscI84cEIWR7nHDs9BtlF2CIOWMWdigdgGEPWj1_BUZtu-kuxog7BCsfpjJOAhlZs1gI6E9EGPgRnMcye4biNUjSBL4sZCLgMpNpIKL0zD3JxyyUMF0ImCwzP-5GNjlRQQIjmUL_6zMKQusHuwaXRr21-uY8xyCbiXS6sEI5D6Z-aJUU84NVIKUd4pXIFLZLS-NW2pQKVdjw0mNGV4rzwsFUEV5pVBFCtM08m5TGKRQtGB2Du1Puvqnfj7o95Jv62Oy6lznhruN5DHHcqfBJVTZ12zba5PtmvVXNR45R3veZ_-mz89CzR22LZl2t9G_0_65vR41xew |
| CitedBy_id | crossref_primary_10_1080_00927872_2022_2026370 crossref_primary_10_1007_s10013_018_00331_5 crossref_primary_10_1142_S0219498823502158 crossref_primary_10_1080_00927872_2023_2286338 crossref_primary_10_2989_16073606_2017_1391354 |
| Cites_doi | 10.1216/RMJ-2012-42-2-425 10.1007/s00009-010-0041-4 10.1017/S0004972714000641 10.1007/s10958-014-1750-1 10.1007/s10958-014-1749-7 10.1080/00927872.2015.1044107 10.4064/cm117-1-2 10.5486/PMD.1980.27.1-2.16 |
| ContentType | Journal Article |
| Copyright | 2016 Australian Mathematical Publishing Association Inc. |
| Copyright_xml | – notice: 2016 Australian Mathematical Publishing Association Inc. |
| DBID | AAYXX CITATION 3V. 7XB 88I 8FE 8FG 8FK ABJCF ABUWG AFKRA AZQEC BENPR BGLVJ CCPQU DWQXO GNUQQ HCIFZ L6V M2P M7S PQEST PQQKQ PQUKI PRINS PTHSS Q9U |
| DOI | 10.1017/S0004972716000435 |
| DatabaseName | CrossRef ProQuest Central (Corporate) ProQuest Central (purchase pre-March 2016) Science Database (Alumni Edition) ProQuest SciTech Collection ProQuest Technology Collection ProQuest Central (Alumni) (purchase pre-March 2016) Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central ProQuest Central Essentials ProQuest Central Technology Collection ProQuest One Community College ProQuest Central Korea ProQuest Central Student SciTech Premium Collection ProQuest Engineering Collection ProQuest Science Journals Engineering Database ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Academic ProQuest One Academic UKI Edition ProQuest Central China Engineering Collection ProQuest Central Basic |
| DatabaseTitle | CrossRef Engineering Database ProQuest Science Journals (Alumni Edition) ProQuest Central Student Technology Collection ProQuest Central Basic ProQuest Central Essentials ProQuest Science Journals ProQuest One Academic Eastern Edition ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College ProQuest Technology Collection ProQuest SciTech Collection ProQuest Central China ProQuest Central ProQuest Engineering Collection ProQuest One Academic UKI Edition ProQuest Central Korea Materials Science & Engineering Collection ProQuest One Academic ProQuest Central (Alumni) Engineering Collection |
| DatabaseTitleList | CrossRef |
| Database_xml | – sequence: 1 dbid: 8FG name: ProQuest Technology Collection url: https://search.proquest.com/technologycollection1 sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| DocumentTitleAlternate | R. R. Andruszkiewicz and M. Woronowicz On the square subgroup of a torsion-free abelian group |
| EISSN | 1755-1633 |
| EndPage | 456 |
| ExternalDocumentID | 10_1017_S0004972716000435 |
| GroupedDBID | --Z -1D -1F -2P -2V -E. -~6 -~N -~X .FH 09C 09E 0E1 0R~ 23N 2WC 4.4 5GY 5VS 6J9 6TJ 6~7 74X 74Y 7~V 88I 9M5 AAAZR AABES AABWE AACJH AAEED AAGFV AAKTX AAMNQ AANRG AARAB AASVR AAUIS AAUKB ABBXD ABBZL ABGDZ ABITZ ABJCF ABJNI ABKKG ABMWE ABMYL ABQTM ABROB ABTAH ABUWG ABVFV ABXAU ABZCX ABZUI ACBMC ACCHT ACETC ACGFO ACGFS ACIMK ACMRT ACQFJ ACREK ACUIJ ACUYZ ACWGA ACYZP ACZBM ACZUX ACZWT ADCGK ADDNB ADFEC ADGEJ ADKIL ADOCW ADOVH ADOVT ADVJH AEBAK AEBPU AEGXH AEHGV AEMTW AENCP AENEX AENGE AEYYC AFFNX AFFUJ AFKQG AFKRA AFLOS AFLVW AFUTZ AGABE AGBYD AGJUD AGLWM AGOOT AHQXX AHRGI AI. AIAGR AIGNW AIHIV AIOIP AISIE AJ7 AJCYY AJPFC AJQAS AKZCZ ALMA_UNASSIGNED_HOLDINGS ALWZO AQJOH ARABE ARZZG ATUCA AUXHV AYIQA AZQEC BBLKV BCGOX BENPR BESQT BGHMG BGLVJ BJBOZ BLZWO BMAJL BQFHP C0O CAG CBIIA CCPQU CCQAD CCUQV CDIZJ CFAFE CFBFF CGQII CHEAL CJCSC COF CS3 DC4 DOHLZ DU5 DWQXO EBS EGQIC EJD ESX GNUQQ HCIFZ HG- HST HZ~ H~9 I.6 I.7 I.9 IH6 IOEEP IOO IS6 I~P J36 J38 J3A JHPGK JQKCU KAFGG KC5 KCGVB KFECR KWQ L98 LHUNA LW7 M-V M2P M7S M7~ M8. NIKVX NMFBF NZEOI O9- OHT OK1 P2P PTHSS PYCCK RAMDC RCA ROL RR0 S10 S6- S6U SAAAG T9M TN5 TWZ UPT UT1 VH1 WFFJZ WQ3 WXU WXY WYP ZCG ZDLDU ZJOSE ZMEZD ZY4 ZYDXJ ~V1 AAYXX ABVZP CITATION CTKSN 3V. 7XB 8FE 8FG 8FK L6V PQEST PQQKQ PQUKI PRINS Q9U |
| ID | FETCH-LOGICAL-c317t-b0ae629c0fb483cf8defca0d1f7c95feda77b613a27de0d222e45942cf6ba3b53 |
| IEDL.DBID | BENPR |
| ISSN | 0004-9727 |
| IngestDate | Thu Oct 10 16:10:19 EDT 2024 Thu Sep 26 19:13:18 EDT 2024 Wed Mar 13 05:47:19 EDT 2024 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 3 |
| Keywords | abelian groups nil-groups primary 20K99 square subgroups secondary 13A99 |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c317t-b0ae629c0fb483cf8defca0d1f7c95feda77b613a27de0d222e45942cf6ba3b53 |
| PQID | 2786995071 |
| PQPubID | 5514771 |
| PageCount | 8 |
| ParticipantIDs | proquest_journals_2786995071 crossref_primary_10_1017_S0004972716000435 cambridge_journals_10_1017_S0004972716000435 |
| PublicationCentury | 2000 |
| PublicationDate | 20161200 2016-12-00 20161201 |
| PublicationDateYYYYMMDD | 2016-12-01 |
| PublicationDate_xml | – month: 12 year: 2016 text: 20161200 |
| PublicationDecade | 2010 |
| PublicationPlace | Cambridge, UK |
| PublicationPlace_xml | – name: Cambridge, UK – name: Cambridge |
| PublicationTitle | Bulletin of the Australian Mathematical Society |
| PublicationTitleAlternate | Bull. Aust. Math. Soc |
| PublicationYear | 2016 |
| Publisher | Cambridge University Press |
| Publisher_xml | – name: Cambridge University Press |
| References | Feigelstock (S0004972716000435_r8) 1983 Stratton (S0004972716000435_r16) 1980; 27 Fuchs (S0004972716000435_r11) 1970 Najafizadeh (S0004972716000435_r15) 2015; 27 Fuchs (S0004972716000435_r10) 1966 Kompantseva (S0004972716000435_r14) 2013; 18 Aghdam (S0004972716000435_r1) 1987; 51 Kompantseva (S0004972716000435_r13) 2012; 17 Feigelstock (S0004972716000435_r9) 2007; 33 S0004972716000435_r18 S0004972716000435_r3 S0004972716000435_r17 S0004972716000435_r2 S0004972716000435_r6 S0004972716000435_r5 Borevich (S0004972716000435_r7) 1966 Fuchs (S0004972716000435_r12) 1973 S0004972716000435_r4 |
| References_xml | – volume: 17 start-page: 63 year: 2012 ident: S0004972716000435_r13 article-title: Absolute nil-ideals of Abelian groups publication-title: Fundam. Prikl. Mat. contributor: fullname: Kompantseva – ident: S0004972716000435_r4 doi: 10.1216/RMJ-2012-42-2-425 – ident: S0004972716000435_r3 doi: 10.1007/s00009-010-0041-4 – volume: 51 start-page: 343 year: 1987 ident: S0004972716000435_r1 article-title: Square subgroup of an Abelian group publication-title: Acta. Sci. Math. contributor: fullname: Aghdam – ident: S0004972716000435_r5 doi: 10.1017/S0004972714000641 – volume: 33 start-page: 641 year: 2007 ident: S0004972716000435_r9 article-title: Additive groups self-injective rings publication-title: Soochow J. Math. contributor: fullname: Feigelstock – volume-title: Infinite Abelian Groups, Vol. 2 year: 1973 ident: S0004972716000435_r12 contributor: fullname: Fuchs – volume-title: Abelian Groups year: 1966 ident: S0004972716000435_r10 contributor: fullname: Fuchs – ident: S0004972716000435_r18 doi: 10.1007/s10958-014-1750-1 – volume-title: Infinite Abelian Groups, Vol. 1 year: 1970 ident: S0004972716000435_r11 contributor: fullname: Fuchs – volume-title: Number Theory year: 1966 ident: S0004972716000435_r7 contributor: fullname: Borevich – ident: S0004972716000435_r17 doi: 10.1007/s10958-014-1749-7 – volume: 18 start-page: 53 year: 2013 ident: S0004972716000435_r14 article-title: Abelian dqt-groups and rings on them publication-title: Fundam. Prikl. Mat. contributor: fullname: Kompantseva – ident: S0004972716000435_r6 doi: 10.1080/00927872.2015.1044107 – volume-title: Additive groups of rings, Vol. 1 year: 1983 ident: S0004972716000435_r8 contributor: fullname: Feigelstock – ident: S0004972716000435_r2 doi: 10.4064/cm117-1-2 – volume: 27 start-page: 127 year: 1980 ident: S0004972716000435_r16 article-title: Abelian groups, nil modulo a subgroup, need not have nil quotient group publication-title: Publ. Math. Debrecen doi: 10.5486/PMD.1980.27.1-2.16 contributor: fullname: Stratton – volume: 27 start-page: 1 year: 2015 ident: S0004972716000435_r15 article-title: On the square submodule of a mixed module publication-title: Gen. Math. Notes contributor: fullname: Najafizadeh |
| SSID | ssj0045528 |
| Score | 2.1434324 |
| Snippet | The first example of a torsion-free abelian group
$(A,+,0)$
such that the quotient group of
$A$
modulo the square subgroup is not a nil-group is indicated (for... |
| SourceID | proquest crossref cambridge |
| SourceType | Aggregation Database Publisher |
| StartPage | 449 |
| SubjectTerms | Group theory Subgroups |
| Title | A TORSION-FREE ABELIAN GROUP EXISTS WHOSE QUOTIENT GROUP MODULO THE SQUARE SUBGROUP IS NOT A NIL-GROUP |
| URI | https://www.cambridge.org/core/product/identifier/S0004972716000435/type/journal_article https://www.proquest.com/docview/2786995071 |
| Volume | 94 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV07T8MwELZoWWBAPEV5yQMTIiJ1_Ign1IJLi9oEmkR0qxzHHttCy__HebVUSEyR7qIMZ_u773yXOwBupbI-RtvzrfKsO04xc7hHqeNJiSnWnHnFX--jgPYT_Dohk-rCbVmVVdaYWAB1Nlf5HfkDYj7lPGcvj4tPJ58alWdXqxEaDbCLbKSAmmC3K4K3cY3FmBBUYrGLHW5ddZ3XLJpG5-zYytq0yIeR390Vtr3UNkgXnqd3CA4qygg75RofgR09Owb7o3W_1eUJMB0Yh-MCF3tjIWCnK2yMHsCiqB6KySCKI_jRDyMB35MwHoggrnSj8DkZhjDuCxjl7NY-km6pGkQwCGPYgcFg6BSiU5D0RPzUd6oJCo6yvGDlpK7UFHHlmhT7njJ-po2SbtY2THFidCYZS61Dl4hl2s0sV9CYcIyUoan0UuKdgeZsPtPnALopp5gaRpSbYRsF-QRJozi1YqRdZFrgfm29aXUOltOyhoxN_xi7Be5qA08XZV-N_16-qpdg8-nNhrj4X30J9izLoWUNyhVorr6-9bVlEqv0BjT83stNtWl-ACSpuxs |
| link.rule.ids | 315,783,787,12779,21402,27938,27939,33387,33758,43614,43819,74371,74638 |
| linkProvider | ProQuest |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV07T8MwELZ4DMCAeIry9MCEiAiJH_GECqQk0CaiSUS3yHHssS20_H8cJ2mpkJgi3VkZzvZ3n33nOwCuudA-Rur9LaqoOyoQtZhLiOVyjgiSjLrm1fsgIkGGXkd41Fy4zZq0yhYTDVCXE1Hdkd851COMVezlYfppVV2jquhq00JjHWwiVzua6qV476VFYoSxUyOxjSymHXUb1TQloyturGX3xETD8O_aCqs-ahWijd_p7YHdhjDCbj3D-2BNjg_AzmBRbXV2CFQXpvHQoGJv6Puw--jrE3oETUo99EdhkibwI4gTH75ncRr6UdroBvFz1o9hGvgwqbit_mSPtSpMYBSnsAujsG8Z0RHIen76FFhN_wRLaFYwtwqbS-IwYasCea5QXimV4HZ5r6hgWMmSU1pod84dWkq71ExBIsyQIxQpuFtg9xhsjCdjeQKgXTCCiKJY2CXSZyAPO1wJRrTYkbajOuB2Yb282QWzvM4go_kfY3fATWvgfFpX1fhv8Hk7BctfL5fD6f_qK7AVpIN-3g-jtzOwrfkOqbNRzsHG_OtbXmhOMS8uzcL5AWJ8u8A |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1LT4QwEG50TYwejM_4tgdPRrIs9EFPZnXBRXdBF4h7I6W0x3V11_9vKeAjJp5IZgiH6Ty-Ml-nAFxyoWuM1PEtqq47KhC1mEuI5XKOCJKMuubU-zgiwww9TPG04T8tGlplmxNNoi5fRfWPvOtQjzBWoZeuamgRT4PgZv5mVTdIVZ3W5jqNVbCmqyKtgtQL7tusjDB26qxsI4vpot12OM346Aona1mPmM4Y_jln4Xe9-p2uTQ0KtsFWAx5hv17tHbAiZ7tgc_w1eXWxB1QfpvHEZMhg4vuwf-vr3XoEDb0e-tMwSRP4MowTHz5ncRr6UdroxvEgG8UwHfowqXCufmS3tSpMYBSnsA-jcGQZ0T7IAj-9G1rNXQqW0AhhaRU2l8RhwlYF8lyhvFIqwe2yp6hgWMmSU1ro0s4dWkq71KhBIsyQIxQpuFtg9wB0Zq8zeQigXTCCiKJY2CXS-yEPO1wJRrTYkbajjsD1l_XyJiIWec0mo_kfYx-Bq9bA-byesPHfy6ftEnx_-ts1jv9XX4B17TP5KIweT8CGhj6kJqacgs7y_UOeaXixLM6N33wCufC_9Q |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+TORSION-FREE+ABELIAN+GROUP+EXISTS+WHOSE+QUOTIENT+GROUP+MODULO+THE+SQUARE+SUBGROUP+IS+NOT+A%C2%A0NIL-GROUP&rft.jtitle=Bulletin+of+the+Australian+Mathematical+Society&rft.au=ANDRUSZKIEWICZ%2C+R.+R.&rft.au=WORONOWICZ%2C+M.&rft.date=2016-12-01&rft.pub=Cambridge+University+Press&rft.issn=0004-9727&rft.eissn=1755-1633&rft.volume=94&rft.issue=3&rft.spage=449&rft.epage=456&rft_id=info:doi/10.1017%2FS0004972716000435&rft.externalDocID=10_1017_S0004972716000435 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0004-9727&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0004-9727&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0004-9727&client=summon |