Left EM rings

Jongwook Baeck

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 3, page 839-867
  • ISSN: 0011-4642

Abstract

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Let R [ x ] be the polynomial ring over a ring R with unity. A polynomial f ( x ) ∈ R [ x ] is referred to as a left annihilating content polynomial (left ACP) if there exist an element r ∈ R and a polynomial g ( x ) ∈ R [ x ] such that f ( x ) = r g ( x ) and g ( x ) is not a right zero-divisor polynomial in R [ x ] . A ring R is referred to as left EM if each polynomial f ( x ) ∈ R [ x ] is a left ACP. We observe the structure of left EM rings with various properties, and study the relationships between the one-sided EM condition and other standard ring theoretic conditions. Moreover, several extensions of EM rings are investigated, including polynomial rings, matrix rings, and Ore localizations.

How to cite

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Baeck, Jongwook. "Left EM rings." Czechoslovak Mathematical Journal 74.3 (2024): 839-867. <http://eudml.org/doc/299297>.

@article{Baeck2024,
abstract = {Let $R[x]$ be the polynomial ring over a ring $R$ with unity. A polynomial $f(x)\in R[x]$ is referred to as a left annihilating content polynomial (left ACP) if there exist an element $r \in R$ and a polynomial $g(x) \in R[x]$ such that $f(x)=rg(x)$ and $g(x)$ is not a right zero-divisor polynomial in $R[x]$. A ring $R$ is referred to as left EM if each polynomial $f(x) \in R[x]$ is a left ACP. We observe the structure of left EM rings with various properties, and study the relationships between the one-sided EM condition and other standard ring theoretic conditions. Moreover, several extensions of EM rings are investigated, including polynomial rings, matrix rings, and Ore localizations.},
author = {Baeck, Jongwook},
journal = {Czechoslovak Mathematical Journal},
keywords = {EM ring; annihilating content polynomial; polynomial ring; uniserial ring; generalized morphic ring; zero-divisor},
language = {eng},
number = {3},
pages = {839-867},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Left EM rings},
url = {http://eudml.org/doc/299297},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Baeck, Jongwook
TI - Left EM rings
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 3
SP - 839
EP - 867
AB - Let $R[x]$ be the polynomial ring over a ring $R$ with unity. A polynomial $f(x)\in R[x]$ is referred to as a left annihilating content polynomial (left ACP) if there exist an element $r \in R$ and a polynomial $g(x) \in R[x]$ such that $f(x)=rg(x)$ and $g(x)$ is not a right zero-divisor polynomial in $R[x]$. A ring $R$ is referred to as left EM if each polynomial $f(x) \in R[x]$ is a left ACP. We observe the structure of left EM rings with various properties, and study the relationships between the one-sided EM condition and other standard ring theoretic conditions. Moreover, several extensions of EM rings are investigated, including polynomial rings, matrix rings, and Ore localizations.
LA - eng
KW - EM ring; annihilating content polynomial; polynomial ring; uniserial ring; generalized morphic ring; zero-divisor
UR - http://eudml.org/doc/299297
ER -

References

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  1. Abuosba, E., Al-Azaizeh, M., Ghanem, M., 10.4134/CKMS.c210439, Commun. Korean Math. Soc. 38 (2023), 69-77. (2023) Zbl1519.13001MR4542621DOI10.4134/CKMS.c210439
  2. Abuosba, E., Ghanem, M., 10.4134/JKMS.j180698, J. Korean Math. Soc. 56 (2019), 1403-1418. (2019) Zbl1427.13024MR3997475DOI10.4134/JKMS.j180698
  3. Agayev, N., Güngöroğlu, G., Harmanci, A., Halicioğlu, S., Abelian modules, Acta Math. Univ. Comen., New Ser. 78 (2009), 235-244. (2009) Zbl1190.16047MR2684191
  4. Anderson, D. D., Abuosba, E., Ghanem, M., 10.1142/S021949882250092X, J. Algebra Appl. 21 (2022), Article ID 2250092, 18 pages. (2022) Zbl1490.13002MR4406639DOI10.1142/S021949882250092X
  5. Anderson, D. D., Anderson, D. F., Markanda, R., 10.1016/0021-8693(85)90096-1, J. Algebra 95 (1985), 96-115. (1985) Zbl0621.13008MR0797658DOI10.1016/0021-8693(85)90096-1
  6. Anderson, D. D., Camillo, V., 10.1080/00927879808826274, Commun. Algebra 26 (1998), 2265-2272. (1998) Zbl0915.13001MR1626606DOI10.1080/00927879808826274
  7. Anderson, D. D., Dumitrescu, T., 10.1081/AGB-120013328, Commun. Algebra 30 (2002), 4407-4416. (2002) Zbl1060.13007MR1936480DOI10.1081/AGB-120013328
  8. Baeck, J., 10.1216/rmj.2021.51.771, Rocky Mt. J. Math. 51 (2021), 771-785. (2021) Zbl1477.16040MR4298828DOI10.1216/rmj.2021.51.771
  9. Baeck, J., 10.1515/math-2022-0545, Open Math. 20 (2022), 1734-1752. (2022) Zbl1508.16008MR4529433DOI10.1515/math-2022-0545
  10. Baeck, J., 10.3934/math.2022673, AIMS Math. 7 (2022), 12106-12122. (2022) MR4431772DOI10.3934/math.2022673
  11. Baeck, J., 10.1007/s13398-023-01514-7, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 118 (2024), Article ID 20, 20 pages. (2024) Zbl07796711MR4662890DOI10.1007/s13398-023-01514-7
  12. Baeck, J., Kim, N. K., Kwak, T. K., Lee, Y., 10.55730/1300-0098.3243, Turk. J. Math. 46 (2022), 1945-1964. (2022) Zbl1515.16034MR4453897DOI10.55730/1300-0098.3243
  13. Baeck, J., Kim, N. K., Lee, Y., Nielsen, P. P., 10.1016/j.jpaa.2020.106432, J. Pure Appl. Algebra 224 (2020), Article ID 106432, 13 pages. (2020) Zbl1465.16037MR4099920DOI10.1016/j.jpaa.2020.106432
  14. Baeck, J., Lee, G., Lim, J. W., 10.11650/tjm.20.2016.7436, Taiwanese J. Math. 20 (2016), 1231-1250. (2016) Zbl1357.16039MR3580293DOI10.11650/tjm.20.2016.7436
  15. Bergman, G. M., 10.1016/0001-8708(78)90010-5, Adv. Math. 29 (1978), 178-218. (1978) Zbl0326.16019MR0506890DOI10.1016/0001-8708(78)90010-5
  16. Bilgin, Z., Reyes, M. L., Tekir, Ü., 10.1080/00927872.2017.1332199, Commun. Algebra 46 (2018), 863-869. (2018) Zbl1410.16026MR3764903DOI10.1080/00927872.2017.1332199
  17. Camillo, V., Nielsen, P. P., 10.1016/j.jpaa.2007.06.010, J. Pure Appl. Algebra 212 (2008), 599-615. (2008) Zbl1162.16021MR2365335DOI10.1016/j.jpaa.2007.06.010
  18. Cannon, G. A., Neuerburg, K. M., 10.35834/mjms/1316032775, Missouri J. Math. Sci. 20 (2008), 165-168. (2008) Zbl1174.16001DOI10.35834/mjms/1316032775
  19. Cui, J., Chen, J., 10.4134/BKMS.2011.48.1.023, Bull. Korean Math. Soc. 48 (2011), 23-33. (2011) Zbl1221.16028MR2778493DOI10.4134/BKMS.2011.48.1.023
  20. Endo, S., 10.1017/S0027763000002129, Nagoya Math. J. 17 (1960), 167-170. (1960) Zbl0117.02203MR0137746DOI10.1017/S0027763000002129
  21. K. R. Goodearl, R. B. Warfield, Jr., 10.1017/CBO9780511841699, London Mathematical Society Student Texts 61. Cambridge University Press, Cambridge (2004). (2004) Zbl1101.16001MR2080008DOI10.1017/CBO9780511841699
  22. Hashemi, E., Estaji, A. AS., Ziembowski, M., 10.1017/S0013091516000407, Proc. Edinb. Math. Soc., II. Ser. 60 (2017), 651-664. (2017) Zbl1405.16036MR3674083DOI10.1017/S0013091516000407
  23. Hong, C. Y., Jeon, Y. C., Kim, N. K., Lee, Y., 10.1080/00927872.2010.480952, Commun. Algebra 39 (2011), 1809-1825. (2011) Zbl1231.16032MR2821508DOI10.1080/00927872.2010.480952
  24. Hong, C. Y., Kim, N. K., Lee, Y., Nielsen, P. P., 10.1016/j.jpaa.2009.01.005, J. Pure Appl. Algebra 213 (2009), 1478-1488. (2009) Zbl1218.16001MR2497591DOI10.1016/j.jpaa.2009.01.005
  25. Hong, C. Y., Kim, N. K., Lee, Y., Ryu, S. J., 10.1016/j.jalgebra.2007.01.042, J. Algebra 315 (2007), 612-628. (2007) Zbl1156.16001MR2351882DOI10.1016/j.jalgebra.2007.01.042
  26. Huckaba, J. A., Keller, J. M., 10.2140/pjm.1979.83.375, Pac. J. Math. 83 (1979), 375-379. (1979) Zbl0388.13001MR0557938DOI10.2140/pjm.1979.83.375
  27. Jordan, D. A., 10.1112/blms/12.3.202, Bull. Lond. Math. Soc. 12 (1980), 202-204. (1980) Zbl0433.16001MR0572101DOI10.1112/blms/12.3.202
  28. Kaplansky, I., Commutative Rings, University of Chicago Press, Chicago (1974). (1974) Zbl0296.13001MR0345945
  29. Kim, N. K., Lee, Y., 10.1006/jabr.1999.8017, J. Algebra 223 (2000), 477-488. (2000) Zbl0957.16018MR1735157DOI10.1006/jabr.1999.8017
  30. Lam, T. Y., 10.1007/978-1-4419-8616-0, Graduate Texts in Mathematics 131. Springer, New York (1991). (1991) Zbl0728.16001MR1125071DOI10.1007/978-1-4419-8616-0
  31. Lam, T. Y., 10.1007/978-1-4612-0525-8, Graduate Texts in Mathematics 189. Springer, New York (1999). (1999) Zbl0911.16001MR1653294DOI10.1007/978-1-4612-0525-8
  32. Lang, S., 10.1007/978-1-4613-0041-0, Graduate Texts in Mathematics 211. Springer, New York (2002). (2002) Zbl0984.00001MR1878556DOI10.1007/978-1-4613-0041-0
  33. Lee, G., Baeck, J., Lim, J. W., 10.11650/tjm/221101, Taiwanese J. Math. 27 (2023), 237-257. (2023) Zbl1529.16018MR4563518DOI10.11650/tjm/221101
  34. Lee, T.-K., Zhou, Y., 10.1081/AGB-120037221, Commun. Algebra 32 (2004), 2287-2299. (2004) Zbl1068.16037MR2100471DOI10.1081/AGB-120037221
  35. Marks, G., Mazurek, R., Ziembowski, M., 10.1017/S0004972709001178, Bull. Aust. Math. Soc. 81 (2010), 361-397. (2010) Zbl1198.16025MR2639852DOI10.1017/S0004972709001178
  36. Mazurek, R., Ziembowski, M., 10.1016/j.jalgebra.2010.11.014, J. Algebra 330 (2011), 130-146. (2011) Zbl1239.16041MR2774621DOI10.1016/j.jalgebra.2010.11.014
  37. Nielsen, P. P., 10.1016/j.jalgebra.2005.10.008, J. Algebra 298 (2006), 134-141. (2006) Zbl1110.16036MR2215121DOI10.1016/j.jalgebra.2005.10.008
  38. Rege, M. B., Chhawchharia, S., 10.3792/pjaa.73.14, Proc. Japan Acad., Ser. A 73 (1997), 14-17. (1997) Zbl0960.16038MR1442245DOI10.3792/pjaa.73.14

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