P-injective group rings

Liang Shen

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 4, page 1103-1109
  • ISSN: 0011-4642

Abstract

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A ring R is called right P-injective if every homomorphism from a principal right ideal of R to R R can be extended to a homomorphism from R R to R R . Let R be a ring and G a group. Based on a result of Nicholson and Yousif, we prove that the group ring RG is right P-injective if and only if (a) R is right P-injective; (b) G is locally finite; and (c) for any finite subgroup H of G and any principal right ideal I of RH , if f ∈ Hom R ( I R , R R ) , then there exists g ∈ Hom R ( RH R , R R ) such that g | I = f . Similarly, we also obtain equivalent characterizations of n -injective group rings and F-injective group rings.

How to cite

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Shen, Liang. "P-injective group rings." Czechoslovak Mathematical Journal 70.4 (2020): 1103-1109. <http://eudml.org/doc/297213>.

@article{Shen2020,
abstract = {A ring $R$ is called right P-injective if every homomorphism from a principal right ideal of $R$ to $R_R$ can be extended to a homomorphism from $R_R$ to $R_R$. Let $R$ be a ring and $G$ a group. Based on a result of Nicholson and Yousif, we prove that the group ring $\{\rm RG\}$ is right P-injective if and only if (a) $R$ is right P-injective; (b) $G$ is locally finite; and (c) for any finite subgroup $H$ of $G$ and any principal right ideal $I$ of $\{\rm RH\}$, if $f\in \{\rm Hom\}_R(I_R, R_R)$, then there exists $g\in \{\rm Hom\}_R(\{\rm RH\}_R, R_R)$ such that $g|_I=f$. Similarly, we also obtain equivalent characterizations of $n$-injective group rings and F-injective group rings.},
author = {Shen, Liang},
journal = {Czechoslovak Mathematical Journal},
keywords = {group ring; P-injective ring; $n$-injective ring; F-injective ring},
language = {eng},
number = {4},
pages = {1103-1109},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {P-injective group rings},
url = {http://eudml.org/doc/297213},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Shen, Liang
TI - P-injective group rings
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 4
SP - 1103
EP - 1109
AB - A ring $R$ is called right P-injective if every homomorphism from a principal right ideal of $R$ to $R_R$ can be extended to a homomorphism from $R_R$ to $R_R$. Let $R$ be a ring and $G$ a group. Based on a result of Nicholson and Yousif, we prove that the group ring ${\rm RG}$ is right P-injective if and only if (a) $R$ is right P-injective; (b) $G$ is locally finite; and (c) for any finite subgroup $H$ of $G$ and any principal right ideal $I$ of ${\rm RH}$, if $f\in {\rm Hom}_R(I_R, R_R)$, then there exists $g\in {\rm Hom}_R({\rm RH}_R, R_R)$ such that $g|_I=f$. Similarly, we also obtain equivalent characterizations of $n$-injective group rings and F-injective group rings.
LA - eng
KW - group ring; P-injective ring; $n$-injective ring; F-injective ring
UR - http://eudml.org/doc/297213
ER -

References

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  2. Farkas, D. R., 10.1090/S0002-9939-1975-0360670-9, Proc. Am. Math. Soc. 48 (1975), 26-28. (1975) Zbl0303.16007MR0360670DOI10.1090/S0002-9939-1975-0360670-9
  3. Ikeda, M., Some generalizations of quasi-Frobenius rings, Osaka Math. J. 3 (1951), 227-239. (1951) Zbl0045.32003MR0046345
  4. Koşan, M. T., Lee, T.-K., Zhou, Y., 10.1007/s10468-012-9388-5, Algebr. Represent. Theory 17 (2014), 87-102. (2014) Zbl1307.16020MR3160714DOI10.1007/s10468-012-9388-5
  5. Nicholson, W. K., Yousif, M. F., 10.1006/jabr.1995.1117, J. Algebra 174 (1995), 77-93. (1995) Zbl0839.16004MR1332860DOI10.1006/jabr.1995.1117
  6. Nicholson, W. K., Yousif, M. F., 10.1017/CBO9780511546525, Cambridge Tracts in Mathematics 158, Cambridge University Press, Cambridge (2003). (2003) Zbl1042.16009MR2003785DOI10.1017/CBO9780511546525
  7. Renault, G., Sur les anneaux des groupes, C. R. Acad. Sci. Paris, Sér. A 273 (1971), 84-87 French. (1971) Zbl0216.06503MR0288189

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